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Diploma in Statistics Design and Analysis of Experiments Lecture 2.21 © 2010 Michael Stuart Design and Analysis of Experiments Lecture 2.2 1.Review of.

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Presentation on theme: "Diploma in Statistics Design and Analysis of Experiments Lecture 2.21 © 2010 Michael Stuart Design and Analysis of Experiments Lecture 2.2 1.Review of."— Presentation transcript:

1 Diploma in Statistics Design and Analysis of Experiments Lecture 2.21 © 2010 Michael Stuart Design and Analysis of Experiments Lecture 2.2 1.Review of Lecture 2.1 and Laboratory 1 2.Homework 2.1.2 3.Introducing the Design Matrix 4.A 2 3 experiment –3 factors each at 2 levels

2 Diploma in Statistics Design and Analysis of Experiments Lecture 2.22 © 2010 Michael Stuart Minute Test: How Much

3 Diploma in Statistics Design and Analysis of Experiments Lecture 2.23 © 2010 Michael Stuart Minute Test: How Fast

4 Diploma in Statistics Design and Analysis of Experiments Lecture 2.24 © 2010 Michael Stuart 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

5 Diploma in Statistics Design and Analysis of Experiments Lecture 2.25 © 2010 Michael Stuart Decomposition of results Effect –0.5 0.7 0.0 –1.2 1.3 Effect –0.4 0.1 0.4

6 Diploma in Statistics Design and Analysis of Experiments Lecture 2.26 © 2010 Michael Stuart Interaction between Factors Case study: Emotional Arousal

7 Diploma in Statistics Design and Analysis of Experiments Lecture 2.27 © 2010 Michael Stuart Interaction between Factors: main effects of pictures vs gender differentiated effects

8 Diploma in Statistics Design and Analysis of Experiments Lecture 2.28 © 2010 Michael Stuart Yield loss experiment

9 Diploma in Statistics Design and Analysis of Experiments Lecture 2.29 © 2010 Michael Stuart Yield loss experiment

10 Diploma in Statistics Design and Analysis of Experiments Lecture 2.210 © 2010 Michael Stuart Yield loss experiment

11 Diploma in Statistics Design and Analysis of Experiments Lecture 2.211 © 2010 Michael Stuart Laboratory 1 Soybean seed failure rates

12 Diploma in Statistics Design and Analysis of Experiments Lecture 2.212 © 2010 Michael Stuart 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.

13 Diploma in Statistics Design and Analysis of Experiments Lecture 2.213 © 2010 Michael Stuart Set up

14 Diploma in Statistics Design and Analysis of Experiments Lecture 2.214 © 2010 Michael Stuart Go to Excel Randomisation

15 Diploma in Statistics Design and Analysis of Experiments Lecture 2.215 © 2010 Michael Stuart Set up: Run order NB: Reset factor levels each time

16 Diploma in Statistics Design and Analysis of Experiments Lecture 2.216 © 2010 Michael Stuart Results (run order)

17 Diploma in Statistics Design and Analysis of Experiments Lecture 2.217 © 2010 Michael Stuart Results (standard order)

18 Diploma in Statistics Design and Analysis of Experiments Lecture 2.218 © 2010 Michael Stuart Analysis (Minitab) Main effects and Interaction plots Pareto plot of effects ANOVA results –with diagnostics Calculation of t-statistic

19 Diploma in Statistics Design and Analysis of Experiments Lecture 2.219 © 2010 Michael Stuart Main Effects and Interactions

20 Diploma in Statistics Design and Analysis of Experiments Lecture 2.220 © 2010 Michael Stuart Bar height = t value (see next slide) Reference line is at critical t value (4 df) df = 7 – 3 = 4

21 Diploma in Statistics Design and Analysis of Experiments Lecture 2.221 © 2010 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 Coef = Effect / 2 SE(Effect) = SE(Coef) x 2

22 Diploma in Statistics Design and Analysis of Experiments Lecture 2.222 © 2010 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

23 Diploma in Statistics Design and Analysis of Experiments Lecture 2.223 © 2010 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

24 Diploma in Statistics Design and Analysis of Experiments Lecture 2.224 © 2010 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

25 Diploma in Statistics Design and Analysis of Experiments Lecture 2.225 © 2010 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

26 Diploma in Statistics Design and Analysis of Experiments Lecture 2.226 © 2010 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

27 Diploma in Statistics Design and Analysis of Experiments Lecture 2.227 © 2010 Michael Stuart Direct Calculation

28 Diploma in Statistics Design and Analysis of Experiments Lecture 2.228 © 2010 Michael Stuart 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 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.

29 Diploma in Statistics Design and Analysis of Experiments Lecture 2.229 © 2010 Michael Stuart Measurements for GC study (EM, Exercise 5.1, pp. 199-200)

30 Diploma in Statistics Design and Analysis of Experiments Lecture 2.230 © 2010 Michael Stuart 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?

31 Diploma in Statistics Design and Analysis of Experiments Lecture 2.231 © 2010 Michael Stuart Organising the data for direct analysis

32 Diploma in Statistics Design and Analysis of Experiments Lecture 2.232 © 2010 Michael Stuart Organising the data for direct analysis s 2 = average(SD 2 ) = ( 2.30 2 + 4.01 2 + 4.92 2 + 3.69 2 ) / 4 = 14.798 s= 3.85 df(s) = sum[df(SD)] = 2 + 2 + 2 + 2 = 8

33 Diploma in Statistics Design and Analysis of Experiments Lecture 2.233 © 2010 Michael Stuart 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

34 Diploma in Statistics Design and Analysis of Experiments Lecture 2.234 © 2010 Michael Stuart Minitab results

35 Diploma in Statistics Design and Analysis of Experiments Lecture 2.235 © 2010 Michael Stuart Minitab results

36 Diploma in Statistics Design and Analysis of Experiments Lecture 2.236 © 2010 Michael Stuart Minitab results

37 Diploma in Statistics Design and Analysis of Experiments Lecture 2.237 © 2010 Michael Stuart 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

38 Diploma in Statistics Design and Analysis of Experiments Lecture 2.238 © 2010 Michael Stuart 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.

39 Diploma in Statistics Design and Analysis of Experiments Lecture 2.239 © 2010 Michael Stuart Minitab results; diagnostics

40 Diploma in Statistics Design and Analysis of Experiments Lecture 2.240 © 2010 Michael Stuart Minitab results; diagnostics

41 Diploma in Statistics Design and Analysis of Experiments Lecture 2.241 © 2010 Michael Stuart Part 4Introducing the design matrix Organising the data for calculation Generic notation

42 Diploma in Statistics Design and Analysis of Experiments Lecture 2.242 © 2010 Michael Stuart 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 2 2 = 4 sets of experimental conditions, as listed.

43 Diploma in Statistics Design and Analysis of Experiments Lecture 2.243 © 2010 Michael Stuart Organising the calculations Main effect of A: average at high A – average at low A = = Main effect of B: average at high B – average at low B = = Columns of design matrix applied to column of means.

44 Diploma in Statistics Design and Analysis of Experiments Lecture 2.244 © 2010 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

45 Diploma in Statistics Design and Analysis of Experiments Lecture 2.245 © 2010 Michael Stuart Calculating interaction effects AB Interaction=½(A effect at high B – A effect at low B) = = The extended design matrix Check:AB = A × B

46 Diploma in Statistics Design and Analysis of Experiments Lecture 2.246 © 2010 Michael Stuart Part 4A 2 3 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

47 Diploma in Statistics Design and Analysis of Experiments Lecture 2.247 © 2010 Michael Stuart A three factor example Design matrix (standard order) Run order for design points (in duplicate)

48 Diploma in Statistics Design and Analysis of Experiments Lecture 2.248 © 2010 Michael Stuart A three factor example Results, in standard order Ref: PilotPlant.xls

49 Diploma in Statistics Design and Analysis of Experiments Lecture 2.249 © 2010 Michael Stuart 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

50 Diploma in Statistics Design and Analysis of Experiments Lecture 2.250 © 2010 Michael Stuart Exercise 2.2.2 Calculate the T, C and K main effects

51 Diploma in Statistics Design and Analysis of Experiments Lecture 2.251 © 2010 Michael Stuart Calculating effects, the extended design matrix Exercise 2.2.3: Complete the missing columns (contrasts). Calculate the TK and TCK interactions

52 Diploma in Statistics Design and Analysis of Experiments Lecture 2.252 © 2010 Michael Stuart Calculating s 2 8 32 2 8 8 2 2 Total 64 s 2 8 s2.828

53 Diploma in Statistics Design and Analysis of Experiments Lecture 2.253 © 2010 Michael Stuart Exercise 2.2.4 Calculate the t-ratio for the T effect and the 3-factor interaction. What conclusions do you draw?

54 Diploma in Statistics Design and Analysis of Experiments Lecture 2.254 © 2010 Michael Stuart 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

55 Diploma in Statistics Design and Analysis of Experiments Lecture 2.255 © 2010 Michael Stuart Minitab analysis

56 Diploma in Statistics Design and Analysis of Experiments Lecture 2.256 © 2010 Michael Stuart Minitab analysis

57 Diploma in Statistics Design and Analysis of Experiments Lecture 2.257 © 2010 Michael Stuart Minitab analysis

58 Diploma in Statistics Design and Analysis of Experiments Lecture 2.258 © 2010 Michael Stuart Minitab analysis

59 Diploma in Statistics Design and Analysis of Experiments Lecture 2.259 © 2010 Michael Stuart Minitab analysis

60 Diploma in Statistics Design and Analysis of Experiments Lecture 2.260 © 2010 Michael Stuart Minitab analysis

61 Diploma in Statistics Design and Analysis of Experiments Lecture 2.261 © 2010 Michael Stuart Minitab diagnostic analysis

62 Diploma in Statistics Design and Analysis of Experiments Lecture 2.262 © 2010 Michael Stuart Minitab diagnostic analysis

63 Diploma in Statistics Design and Analysis of Experiments Lecture 2.263 © 2010 Michael Stuart Homework 2.2.1 An experiment was run to assess the effects of three factors on the life of a cutting tool A:Cutting speed B:Tool geometry C:Cutting angle. The full 2 3 design was replicated three times. The results are shown in the next slide and are available in Excel file Tool Life.xls. Carry out a full analysis and report. Ref: Tool Life.xls

64 Diploma in Statistics Design and Analysis of Experiments Lecture 2.264 © 2010 Michael Stuart Homework 2.2.1 Homework 2.2.2: Web Exercises See also Homework 1.2.2 Ref: Hardness.xls

65 Diploma in Statistics Design and Analysis of Experiments Lecture 2.265 © 2010 Michael Stuart Reading EM §5.3, §5.4, §5.6 DCM §6-2, §6-3 to p.218, §6.5 to p. 235

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