Design and Analysis of Experiments Lecture 2 2

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 23 experiment – 3 factors each at 2 levels Diploma in Statistics Design and Analysis of Experiments 1

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

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

Yield Loss Experiment: Blends in Randomised Blocks General Linear Model: Loss, per cent versus Blend, Block Analysis of Variance for Loss, %, Source Blend Block Error Total DF 4 2 8 14 SS 11. 5560 1. 6480 6. 9920 Diploma in Statistics Design and Analysis of Experiments MS 2. 8890 0. 8240 0. 8740 F 3. 31 0. 94 4 P 0. 071 0. 429

Decomposition of results Effect – 0. 5 0. 7 0. 0 – 1. 2 1. 3 Effect – 0. 4 Diploma in Statistics Design and Analysis of Experiments 0. 1 0. 4 5

Interaction between Factors Case study: Emotional Arousal Diploma in Statistics Design and Analysis of Experiments 6

Interaction between Factors: main effects of pictures vs gender differentiated effects Diploma in Statistics Design and Analysis of Experiments 7

Yield loss experiment 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

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

A 22 experiment Project: optimisation of a chemical process yield Factors (with levels): operating temperature (Low, High) catalyst (C 1, C 2) Design: Process run at all four possible combinations of factor levels, in duplicate, in random order. Diploma in Statistics Design and Analysis of Experiments 12

Set up Diploma in Statistics Design and Analysis of Experiments 13

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

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

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

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

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 18

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

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 20

Minitab DOE Analyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Constant Temperature Catalyst Temperature*Catalyst Effect 23. 0000 1. 5000 10. 0000 Coef 64. 2500 11. 5000 0. 7500 5. 0000 SE Coef 1. 311 S = 3. 70810 Coef = Effect / 2 SE(Effect) = SE(Coef) x 2 Diploma in Statistics Design and Analysis of Experiments 21 T 49. 01 8. 77 0. 57 3. 81 P 0. 000 0. 001 0. 598 0. 019

Minitab DOE Analyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Constant Temperature Catalyst Temperature*Catalyst S = 3. 70810 Effect Coef 64. 2500 11. 5000 0. 7500 5. 0000 23. 0000 1. 5000 10. 0000 R-Sq = 95. 83% SE Coef 1. 311 T 49. 01 8. 77 0. 57 3. 81 P 0. 000 0. 001 0. 598 0. 019 R-Sq(adj) = 92. 69% Analysis of Variance for Yield (coded units) Source Main Effects 2 -Way Interactions Residual Error Pure Error Total DF 2 1 4 4 7 Seq SS 1062. 50 200. 00 55. 00 1317. 50 Diploma in Statistics Design and Analysis of Experiments Adj SS 1062. 50 200. 00 55. 00 Adj MS 531. 25 200. 00 13. 75 22 F 38. 64 14. 55 P 0. 002 0. 019

Minitab DOE Analyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Constant Temperature Catalyst Temperature*Catalyst S = 3. 70810 Effect Coef 64. 2500 11. 5000 0. 7500 5. 0000 23. 0000 1. 5000 10. 0000 R-Sq = 95. 83% SE Coef 1. 311 T 49. 01 8. 77 0. 57 3. 81 P 0. 000 0. 001 0. 598 0. 019 R-Sq(adj) = 92. 69% Analysis of Variance for Yield (coded units) Source Main Effects 2 -Way Interactions Residual Error Pure Error Total DF 2 1 4 4 7 Seq SS 1062. 50 200. 00 55. 00 1317. 50 Diploma in Statistics Design and Analysis of Experiments Adj SS 1062. 50 200. 00 55. 00 Adj MS 531. 25 200. 00 13. 75 23 F 38. 64 14. 55 P 0. 002 0. 019

Minitab DOE Analyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Constant Temperature Catalyst Temperature*Catalyst S = 3. 70810 Effect Coef 64. 2500 11. 5000 0. 7500 5. 0000 23. 0000 1. 5000 10. 0000 R-Sq = 95. 83% SE Coef 1. 311 T 49. 01 8. 77 0. 57 3. 81 P 0. 000 0. 001 0. 598 0. 019 R-Sq(adj) = 92. 69% Analysis of Variance for Yield (coded units) Source Main Effects 2 -Way Interactions Residual Error Pure Error Total DF 2 1 4 4 7 Seq SS 1062. 50 200. 00 55. 00 1317. 50 Diploma in Statistics Design and Analysis of Experiments Adj SS 1062. 50 200. 00 55. 00 Adj MS 531. 25 200. 00 13. 75 24 F 38. 64 14. 55 P 0. 002 0. 019

Minitab DOE Analyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Constant Temperature Catalyst Temperature*Catalyst S = 3. 70810 Effect Coef 64. 2500 11. 5000 0. 7500 5. 0000 23. 0000 1. 5000 10. 0000 R-Sq = 95. 83% SE Coef 1. 311 T 49. 01 8. 77 0. 57 3. 81 P 0. 000 0. 001 0. 598 0. 019 R-Sq(adj) = 92. 69% Analysis of Variance for Yield (coded units) Source Main Effects 2 -Way Interactions Residual Error Pure Error Total DF 2 1 4 4 7 Seq SS 1062. 50 200. 00 55. 00 1317. 50 Diploma in Statistics Design and Analysis of Experiments Adj SS 1062. 50 200. 00 55. 00 Adj MS 531. 25 200. 00 13. 75 25 F 38. 64 14. 55 P 0. 002 0. 019

Minitab DOE Analyze Factorial Design Estimated Effects and Coefficients for Yield (coded units) Term Constant Temperature Catalyst Temperature*Catalyst S = 3. 70810 Effect Coef 64. 2500 11. 5000 0. 7500 5. 0000 23. 0000 1. 5000 10. 0000 R-Sq = 95. 83% SE Coef 1. 311 T 49. 01 8. 77 0. 57 3. 81 P 0. 000 0. 001 0. 598 0. 019 R-Sq(adj) = 92. 69% Analysis of Variance for Yield (coded units) Source Main Effects 2 -Way Interactions Residual Error Pure Error Total DF 2 1 4 4 7 Seq SS 1062. 50 200. 00 55. 00 1317. 50 Diploma in Statistics Design and Analysis of Experiments Adj SS 1062. 50 200. 00 55. 00 Adj MS 531. 25 200. 00 13. 75 26 F 38. 64 14. 55 P 0. 002 0. 019

Direct Calculation Diploma in Statistics Design and Analysis of Experiments 27

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 28

Measurements for GC study (EM, Exercise 5. 1, pp. 199 -200) Diploma in Statistics Design and Analysis of Experiments 29

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 30

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

Organising the data for direct analysis s 2 = average(SD 2) = ( 2. 302 + 4. 012 + 4. 922 + 3. 692 ) / 4 = 14. 798 s df(s) = 3. 85 = sum[df(SD)] =2+2+2+2 =8 Diploma in Statistics Design and Analysis of Experiments 32

Minitab results Estimated Effects for Measurements Term Flow rate Volume Flow rate*Volume Effect -19. 233 98. 233 8. 767 SE 2. 222 T P -8. 66 44. 21 3. 95 0. 000 0. 004 S = 3. 84816 Diploma in Statistics Design and Analysis of Experiments 33

Minitab results 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

More Minitab results Means for Peak area Mean SE Mean 88. 10 68. 87 1. 571 Volume 100 29. 37 127. 60 1. 571 Flow 200 400 43. 37 15. 37 132. 83 122. 37 2. 222 Flow rate 200 400 rate*Volume 100 200 Diploma in Statistics Design and Analysis of Experiments 37

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 38

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

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

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

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 42

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

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 44

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 45

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 46

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

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

Minitab analysis Estimated Effects for Yield Term T C K T*C T*K C*K T*C*K Effect 23. 0 -5. 0 1. 5 10. 0 0. 5 SE 1. 414 1. 414 T 16. 26 -3. 54 1. 06 7. 07 0. 00 0. 35 P 0. 000 0. 008 0. 320 0. 000 1. 000 0. 733 S = 2. 82843 Diploma in Statistics Design and Analysis of Experiments 49

Exercise 2. 2. 2 Calculate the T, C and K main effects Diploma in Statistics Design and Analysis of Experiments 50

Calculating effects, the extended design matrix Exercise 2. 2. 3: Complete the missing columns (contrasts). Calculate the TK and TCK interactions Diploma in Statistics Design and Analysis of Experiments 51

Calculating s 2 8 32 2 8 8 2 2 Total Diploma in Statistics Design and Analysis of Experiments 64 s 2 8 s 2. 828 52

Exercise 2. 2. 4 Calculate the t-ratio for the T effect and the 3 -factor interaction. What conclusions do you draw? Diploma in Statistics Design and Analysis of Experiments 53

Minitab analysis Estimated Effects for Yield Term T C K T*C T*K C*K T*C*K Effect 23. 0 -5. 0 1. 5 10. 0 0. 5 SE 1. 414 1. 414 T 16. 26 -3. 54 1. 06 7. 07 0. 00 0. 35 P 0. 000 0. 008 0. 320 0. 000 1. 000 0. 733 S = 2. 82843 Diploma in Statistics Design and Analysis of Experiments 54

Minitab analysis Diploma in Statistics Design and Analysis of Experiments 55

Minitab analysis Diploma in Statistics Design and Analysis of Experiments 56

Minitab analysis Diploma in Statistics Design and Analysis of Experiments 57

Minitab analysis Diploma in Statistics Design and Analysis of Experiments 58

Minitab analysis Diploma in Statistics Design and Analysis of Experiments 59

Minitab analysis Diploma in Statistics Design and Analysis of Experiments 60

Minitab diagnostic analysis Diploma in Statistics Design and Analysis of Experiments 61

Minitab diagnostic analysis Diploma in Statistics Design and Analysis of Experiments 62

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 23 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. Diploma in Statistics Design and Analysis of Experiments Ref: Tool Life. xls 63

Homework 2. 2. 1 Homework 2. 2. 2: Web Exercises Ref: Hardness. xls See also Homework 1. 2. 2 Diploma in Statistics Design and Analysis of Experiments 64

Reading EM § 5. 3, § 5. 4, § 5. 6 DCM § 6 -2, § 6 -3 to p. 218, § 6. 5 to p. 235 Diploma in Statistics Design and Analysis of Experiments 65