1. Lecture 2.21  2016 Michael Stuart Design and Analysis of Experiments Lecture 2.2 1.Review Lecture 2.1 –Minute test –Why block? –Deleted residuals 2.Interaction 3.Random Block Effects 4.Introduction to 2-level factorial designs –a 2 2 experiment –introducing the Design Matrix Postgraduate Certificate in Statistics Design and Analysis of Experiments

2. Lecture 2.22  2016 Michael Stuart How Much Postgraduate Certificate in Statistics Design and Analysis of Experiments

3. Lecture 2.23  2016 Michael Stuart How Fast Postgraduate Certificate in Statistics Design and Analysis of Experiments

4. Lecture 2.24  2016 Michael Stuart Why block? Blocking is useful when there are known external factors (covariates) that affect variation between plots. Blocking reduces bias arising due to block effects disproportionately affecting factor effects due to levels disproportionally allocated to blocks. Neighbouring plots are likely to be more homogeneous than separated plots, so that –blocking reduces variation in results when treatments are compared within blocks –(and increases precision when results are combined across blocks). Postgraduate Certificate in Statistics Design and Analysis of Experiments

5. Lecture 2.25  2016 Michael Stuart Deleted residuals Minitab does this automatically for all cases! They are used to allow each case to be assessed using a criterion not affected by the case. The residuals are not deleted, it is the case that is deleted while the corresponding "deleted residual is calculated Simple linear regression illustrates: Postgraduate Certificate in Statistics Design and Analysis of Experiments

6. Lecture 2.26  2016 Michael Stuart Scatterplot Postgraduate Certificate in Statistics Design and Analysis of Experiments

7. Lecture 2.27  2016 Michael Stuart Scatterplot Postgraduate Certificate in Statistics Design and Analysis of Experiments

8. Lecture 2.28  2016 Michael Stuart Scatterplot Postgraduate Certificate in Statistics Design and Analysis of Experiments

9. Lecture 2.29  2016 Michael Stuart Deleted residual Given an exceptional case, deleted residual> residual using all the data deleted s< s using all the data deleted standardised residual >> standardised residual using all the data Using deleted residuals accentuates exceptional cases Postgraduate Certificate in Statistics Design and Analysis of Experiments

10. Lecture 2.210  2016 Michael Stuart Residuals Postgraduate Certificate in Statistics Design and Analysis of Experiments

11. Lecture 2.211  2016 Michael Stuart Design and Analysis of Experiments Lecture 2.2 1.Review Lecture 2.1 –Minute test –Why block? –Deleted residuals 2.Interaction 3.Random Block Effects 4.Introduction to 2-level factorial designs –a 2 2 experiment –introducing the Design Matrix Postgraduate Certificate in Statistics Design and Analysis of Experiments

12. Lecture 2.212  2016 Michael Stuart Multi-factor designs reveal interaction Pressure Temperature High Low 65 75 70 60 Postgraduate Certificate in Statistics Design and Analysis of Experiments

13. Lecture 2.213  2016 Michael Stuart Interaction defined Factors interact when the effect of changing one factor depends on the level of the other. Interaction displayed Postgraduate Certificate in Statistics Design and Analysis of Experiments

14. Lecture 2.214  2016 Michael Stuart Iron-deficiency anemia Contributory factors: –cooking pot type Aluminium (A), Clay (C) and Iron (I) –food type Meat (M), Legumes (L) and Vegetables (V) Postgraduate Certificate in Statistics Design and Analysis of Experiments

15. Lecture 2.215  2016 Michael Stuart Interaction Postgraduate Certificate in Statistics Design and Analysis of Experiments

16. Lecture 2.216  2016 Michael Stuart Interaction Postgraduate Certificate in Statistics Design and Analysis of Experiments LegumesVegetableMeat Aluminium2.331.232.06 Change effect0.140.230.12 Clay2.471.462.18 Change effect1.201.332.50 Iron3.672.794.68

17. Lecture 2.217  2016 Michael Stuart Interaction Postgraduate Certificate in Statistics Design and Analysis of Experiments LegumesVegetableMeat Aluminium2.331.232.06 Change effect0.140.230.12 Clay2.471.462.18 Change effect1.201.332.50 Iron3.672.794.68

18. Lecture 2.218  2016 Michael Stuart Two 2-level factors Pressure Temperature High Low 65 75 70 60 Pressure effect Low T: 60 – 65 = –5 High T: 75 – 70 = +5 Diff:5 – (–5) = 10 Temperature effect Low P: 70 – 65 = 5 High P: 75 – 60 = 15 Diff:15 – 5 = 10 Postgraduate Certificate in Statistics Design and Analysis of Experiments

19. Lecture 2.219  2016 Michael Stuart Model for analysis Iron content includes –a contribution for each food type plus –a contribution for each pot type plus –a contribution for each food type / pot type combination plus –a contribution due to chance variation Postgraduate Certificate in Statistics Design and Analysis of Experiments

20. Lecture 2.220  2016 Michael Stuart Model for analysis Y =  +  +  +  where  is the overall mean,  is the food effect, above or below the mean, depending on which food type is used,  is the pot effect, above or below the mean, depending on which pot type is involved  is the food/pot interaction effect, depending on which food type / pot type combination is used  represents chance variation Postgraduate Certificate in Statistics Design and Analysis of Experiments

21. Lecture 2.221  2016 Michael Stuart Estimating the model Food Type Pot Means Pot Main Effects MLV A2.12.31.21.91.9 – 2.5 = – 0.6 Pot Type C2.22.51.52.02.0 – 2.5 = – 0.5 I4.73.72.83.73.7 – 2.5 = + 1.2 Food Means 3.02.81.82.5 Food Main Effects 3.0 – 2.5 = +0.5 2.8 – 2.5 = +0.3 1.8 – 2.5 = – 0.7 Postgraduate Certificate in Statistics Design and Analysis of Experiments

22. Lecture 2.222  2016 Michael Stuart Estimating the model Food Type Pot Means Pot Main Effects MLV A2.12.31.21.91.9 – 2.5 = – 0.6 Pot Type C2.22.51.52.02.0 – 2.5 = – 0.5 I4.73.72.83.73.7 – 2.5 = + 1.2 Food Means 3.02.81.82.5 Food Main Effects 3.0 – 2.5 = +0.5 2.8 – 2.5 = +0.3 1.8 – 2.5 = – 0.7 Postgraduate Certificate in Statistics Design and Analysis of Experiments

23. Lecture 2.223  2016 Michael Stuart Interaction effects Postgraduate Certificate in Statistics Design and Analysis of Experiments MLV Pot Effects A2.12.31.2-0.7 C2.22.51.5-0.5 I4.73.72.81.2 Food Effects 0.40.3-0.72.5 Interaction Effects -0.20.20.1 -0.30.20.1 0.5-0.3-0.2

24. Lecture 2.224  2016 Michael Stuart Interaction effects Postgraduate Certificate in Statistics Design and Analysis of Experiments MLV Pot Effects A2.12.31.2-0.7 C2.22.51.5-0.5 I4.73.72.81.2 Food Effects 0.40.3-0.72.5 Interaction Effects -0.20.20.1 -0.30.20.1 0.5-0.3-0.2

25. Lecture 2.225  2016 Michael Stuart Estimating  Calculate s from each cell, based on 4 – 1 = 3 df, Estimate is average across all 9 cells, with 9 x 3 = 27 df Postgraduate Certificate in Statistics Design and Analysis of Experiments

26. Lecture 2.226  2016 Michael Stuart Analysis of Variance SS(Total) = SS(Pot effects) + SS(Food effects) + SS(Interaction effects) + SS(Error) Source DF SS MS F-Value P-Value Pot 2 24.9 12.4 92.3 0.000 Food 2 9.3 4.6 34.5 0.000 Pot*Food 4 2.6 0.66 4.9 0.004 Error 27 3.6 0.13 Total 35 40.5 Postgraduate Certificate in Statistics Design and Analysis of Experiments

27. Lecture 2.227  2016 Michael Stuart Recall:Case Study Reducing yield loss in a chemical process Process: chemicals blended, filtered and dried 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) Postgraduate Certificate in Statistics Design and Analysis of Experiments

28. Lecture 2.228  2016 Michael Stuart Results Ref: BlendLoss.xls Postgraduate Certificate in Statistics Design and Analysis of Experiments

29. Lecture 2.229  2016 Michael Stuart Initial data analysis Little variation between blocks More variation between blends Disturbing interaction pattern; see later Postgraduate Certificate in Statistics Design and Analysis of Experiments

30. Lecture 2.230  2016 Michael Stuart Analysis of Variance Blend Loss analysis model included Blend effects + Block effects + Chance variation, –NO INTERACTION EFFECTS Analysis of Variance for Loss 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 Postgraduate Certificate in Statistics Design and Analysis of Experiments

31. Lecture 2.231  2016 Michael Stuart Include interaction in model? Analysis of Variance for Loss Source DF Adj SS Adj MS F P Blend 4 11.5560 2.8890 ** Block 2 1.6480 0.8240 ** Blend*Block 8 6.9920 0.8740 ** Error 0 * * * Total 14 20.1960 Postgraduate Certificate in Statistics Design and Analysis of Experiments

32. Lecture 2.232  2016 Michael Stuart ANOVA with no replication Recall F-test logic: –MS(Error) ≈  2 –MS(Effect) ≈  2 + effect contribution –F = MS(Effect) / MS(Error) ≈ 1 if effect absent, >>1 if effect present No replication? use MS(Interaction) as MS(Error) If Block by Treatment interaction is absent, –OK If Block by Treatment interaction is present, –conservative test Postgraduate Certificate in Statistics Design and Analysis of Experiments

33. Lecture 2.233  2016 Michael Stuart Fitted values show no interaction. Postgraduate Certificate in Statistics Design and Analysis of Experiments Recall :Estimating the model

34. Lecture 2.234  2016 Michael Stuart Classwork 2.1.2 Calculate fitted values Postgraduate Certificate in Statistics Design and Analysis of Experiments 17.5 +

35. Lecture 2.235  2016 Michael Stuart Classwork 2.1.2 (cont'd) Make a Block profile plot Postgraduate Certificate in Statistics Design and Analysis of Experiments

36. Lecture 2.236  2016 Michael Stuart Fitted values; NO INTERACTION Postgraduate Certificate in Statistics Design and Analysis of Experiments

37. Lecture 2.237  2016 Michael Stuart Interaction? Blend x Block interaction? no general test without replication Postgraduate Certificate in Statistics Design and Analysis of Experiments

38. Lecture 2.238  2016 Michael Stuart Design and Analysis of Experiments Lecture 2.2 1.Review Lecture 2.1 –Minute test –Why block? –Deleted residuals 2.Interaction 3.Random Block Effects 4.Introduction to 2-level factorial designs –a 2 2 experiment –introducing the Design Matrix Postgraduate Certificate in Statistics Design and Analysis of Experiments

39. Lecture 2.239  2016 Michael Stuart Part 3Random block effects Contribution of blend effect is predictable, depends on the known makeup of each blend Contribution of block effect is not predictable, depends on current conditions at run time. Convention: –Blend effect is fixed, –Block effect is random  A,  B,  C,  D,  E are fixed but unknown,  I,  II,  III are random numbers Assumption:   N( 0,  B ) Postgraduate Certificate in Statistics Design and Analysis of Experiments

40. Lecture 2.240  2016 Michael Stuart Random block effects Recall F-test logic: –MS(Error) ≈  2 –MS(Effect) ≈  2 + effect contribution –F = MS(Effect) / MS(Error) ≈ 1 if effect absent, >>1 if effect present For Blend Effect, effect contribution= For Block Effect, effect contribution= No effect on logic of F-test Postgraduate Certificate in Statistics Design and Analysis of Experiments

41. Lecture 2.241  2016 Michael Stuart Minitab analysis Analysis of Variance for Loss, using Adjusted SS for Tests Source DF Adj SS Adj MS F-Value P-Value Block 2 1.648 0.8240 0.94 0.429 Blend 4 11.556 2.8890 3.31 0.071 Error 8 6.992 0.8740 Total 14 20.196 Expected Mean Square Source for Each Term 1 Block (3) + 5.0000 (1) 2 Blend (3) + Q[2] 3 Error (3) Ref: DCM, p. 125, p. 133 Postgraduate Certificate in Statistics Design and Analysis of Experiments

42. Lecture 2.242  2016 Michael Stuart Design and Analysis of Experiments Lecture 2.2 1.Review Lecture 2.1 –Minute test –Why block? –Deleted residuals 2.Interaction 3.Random Block Effects 4.Introduction to 2-level factorial designs –a 2 2 experiment –introducing the Design Matrix Postgraduate Certificate in Statistics Design and Analysis of Experiments

43. Lecture 2.243  2016 Michael Stuart Part 4 Introduction 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. Postgraduate Certificate in Statistics Design and Analysis of Experiments

44. Lecture 2.244  2016 Michael Stuart Design set up Postgraduate Certificate in Statistics Design and Analysis of Experiments

45. Lecture 2.245  2016 Michael Stuart Go to Excel Randomisation Postgraduate Certificate in Statistics Design and Analysis of Experiments

46. Lecture 2.246  2016 Michael Stuart Design set up: Run order NB: Reset factor levels each time Postgraduate Certificate in Statistics Design and Analysis of Experiments

47. Lecture 2.247  2016 Michael Stuart Classwork 2.2.3 What were the experimental units factors factor levels treatments response blocks allocation procedure Postgraduate Certificate in Statistics Design and Analysis of Experiments

48. Lecture 2.248  2016 Michael Stuart Results (run order) Postgraduate Certificate in Statistics Design and Analysis of Experiments

49. Lecture 2.249  2016 Michael Stuart Results (standard order) Postgraduate Certificate in Statistics Design and Analysis of Experiments

50. Lecture 2.250  2016 Michael Stuart Analysis (Minitab) Main effects and Interaction plots ANOVA results –with diagnostics Calculation of t-statistics Postgraduate Certificate in Statistics Design and Analysis of Experiments

51. Lecture 2.251  2016 Michael Stuart Main Effects and Interactions Postgraduate Certificate in Statistics Design and Analysis of Experiments

52. Lecture 2.252  2016 Michael Stuart Minitab DOE command; Estimated Effects and Coefficients for Yield Term Effect Coef SE Coef T P Constant 64.25 1.31 49.01 0.000 Temperature 23.0 11.50 1.31 8.77 0.001 Catalyst 1.5 0.75 1.31 0.57 0.598 Temperature*Catalyst 10.0 5.00 1.31 3.81 0.019 S = 3.70810 Effect = Coef x 2 SE(Effect) = SE(Coef) x 2 Analyze Factorial Design subcommand Postgraduate Certificate in Statistics Design and Analysis of Experiments

53. Lecture 2.253  2016 Michael Stuart 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 Postgraduate Certificate in Statistics Design and Analysis of Experiments

54. Lecture 2.254  2016 Michael Stuart 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 Postgraduate Certificate in Statistics Design and Analysis of Experiments

55. Lecture 2.255  2016 Michael Stuart 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 Postgraduate Certificate in Statistics Design and Analysis of Experiments

56. Lecture 2.256  2016 Michael Stuart 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, Section 5.10, p.188 “The standard form of the ‘analysis of variance’ does not seem to me to be useful for 2 n data. Daniel (1976), Section 7.1, p.128 Postgraduate Certificate in Statistics Design and Analysis of Experiments

57. Lecture 2.257  2016 Michael Stuart Diagnostic Plots Postgraduate Certificate in Statistics Design and Analysis of Experiments

58. Lecture 2.258  2016 Michael Stuart Direct Calculation Postgraduate Certificate in Statistics Design and Analysis of Experiments

59. Lecture 2.259  2016 Michael Stuart Classwork 2.2.2 Calculate a confidence interval for the Temperature effect. All effects may be estimated and tested in this way. Homework 2.2.1 Test the statistical significance of and calculate confidence intervals for the Catalyst effect and the Temperature by Catalyst interaction effect. Postgraduate Certificate in Statistics Design and Analysis of Experiments

60. Lecture 2.260  2016 Michael Stuart 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 Postgraduate Certificate in Statistics Design and Analysis of Experiments

61. Lecture 2.261  2016 Michael Stuart 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 ) Postgraduate Certificate in Statistics Design and Analysis of Experiments

62. Lecture 2.262  2016 Michael Stuart Exercise 2.2.1 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. Postgraduate Certificate in Statistics Design and Analysis of Experiments

63. Lecture 2.263  2016 Michael Stuart Measurements for GC study (EM, Exercise 5.1, pp. 199-200) Postgraduate Certificate in Statistics Design and Analysis of Experiments

64. Lecture 2.264  2016 Michael Stuart Introducing the Design Matrix Postgraduate Certificate in Statistics Design and Analysis of Experiments

65. Lecture 2.265  2016 Michael Stuart Design Matrix Postgraduate Certificate in Statistics Design and Analysis of Experiments

66. Lecture 2.266  2016 Michael Stuart Design Matrix with Y’s Postgraduate Certificate in Statistics Design and Analysis of Experiments

67. Lecture 2.267  2016 Michael Stuart Design Matrix with Data Postgraduate Certificate in Statistics Design and Analysis of Experiments

68. Lecture 2.268  2016 Michael Stuart Augmented Design Matrix with Y’s Postgraduate Certificate in Statistics Design and Analysis of Experiments

69. Lecture 2.269  2016 Michael Stuart Augmented Design Matrix with Data Calculate effects as Mean(+) – Mean(–) Postgraduate Certificate in Statistics Design and Analysis of Experiments

70. Lecture 2.270  2016 Michael Stuart 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 Postgraduate Certificate in Statistics Design and Analysis of Experiments

71. Lecture 2.271  2016 Michael Stuart Reading EM §5.3 DCM §6-2, §6-2 Postgraduate Certificate in Statistics Design and Analysis of Experiments