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Design and Analysis of Experiments Lecture 6.1

Design and Analysis of Experiments Lecture 6.1. Review of split unit experiments Review of Laboratory 2 Review of special topics (part). Minute Test: How Much. Minute Test: How Fast. Split units experiments. arise when one set of treatment factors is applied to experimental units,

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Design and Analysis of Experiments Lecture 6.1

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  1. Design and Analysis of ExperimentsLecture 6.1 • Review of split unit experiments • Review of Laboratory 2 • Review of special topics (part) 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. Split units experiments arise when • one set of treatment factors is applied to experimental units, • a second set of factors is applied to sub units of these experimental units. originated in agriculture where they are referred to as split plot designs. "Most industrial experiments are ... split plot in their design.“ C. Daniel (1976) p. 175 Diploma in Statistics Design and Analysis of Experiments

  5. Reasons for using split units • Adding another factor after the experiment started • Changing one factor is • more difficult • more expensive • more time consuming • than changing others • Some factors require better precision than others Diploma in Statistics Design and Analysis of Experiments

  6. Recognising Plot and Treatment Structure Units Blocks Whole units Subunits Factor Whole unit Treatment Subunit Treatment ANOVA MS(Blocks) MS(WTreatments) MS(Whole units) MS(B x WT) MS(STreatments) MS(Interactions) MS(Subunits) Diploma in Statistics Design and Analysis of Experiments

  7. Illustration: Water resistance of wood stains Diploma in Statistics Design and Analysis of Experiments

  8. Plot structure, Treatment structure Factor Pretreatment Stain Units Boards Panels 24 subunits (panels) nested in 6 whole units (boards). 2 pretreatments allocated to whole units. 4 stains allocated to subunits within whole units. Diploma in Statistics Design and Analysis of Experiments

  9. Assessing variation • Variation between boards due to • chance • Pretreatments? • Variation between panels due to • chance • stains? • pretreatment by stain interaction? Diploma in Statistics Design and Analysis of Experiments

  10. ANOVA MS(Pretreatment) MS(Boards) MS(Stain) MS(Interaction) MS(Panels) Analysis of Variance Factor Pretreatment Stain Units Boards Panels Minitab model • Pretreatment Board(Pretreatment) • Stain Pretreatment*Stain Diploma in Statistics Design and Analysis of Experiments

  11. Analysis of Variance Source DF SS MS F P Pretreatment 1 782.04 782.04 4.03 0.115 Board(Pretreatment) 4 775.36 193.84 15.25 0.000 Stain 3 266.00 88.67 6.98 0.006 Pretreatment*Stain 3 62.79 20.93 1.65 0.231 Error 12 152.52 12.71 Total 23 2038.72 Diploma in Statistics Design and Analysis of Experiments

  12. Justifying the ANOVA Source Expected Mean Square 1 Pretreat (5) + 4.0000 (2) + Q[1] 2 Board(Pretreat) (5) + 4.0000 (2) 3 Stain (5) + Q[3] 4 Pretreat*Stain (5) + Q[4] 5 Error (5) Alternative notation: Source Expected Mean Square 1 Pretreat + 4 + Pretreatment effect 2 Board(Pretreat) + 4 3 Stain + Stain effect 4 Pretreat*Stain + interaction effect 5 Error Diploma in Statistics Design and Analysis of Experiments

  13. Extending the unit structure Suppose the 6 boards were in 3 blocks of 2 e.g. 2 boards selected from 3 production runs, e.g. 2 boards treated on 3 successive days Diploma in Statistics Design and Analysis of Experiments

  14. Expanded Unit and Treatment Structure Units Blocks Boards Panels Factor Pretreatment Stain ANOVA MS(Blocks) MS(Pretreatment) MS(Boards) MS(B x PT) MS(Stain) MS(Interactions) MS(Panels) Diploma in Statistics Design and Analysis of Experiments

  15. Analysis of Variance Minitab model Block Pretreatment Block*Pretreatment Stain Block*Stain Pretreatment*Stain Source DF SS MS F P Block 2 376.99 188.49 0.95 0.514 Pretreatment 1 782.04 782.04 3.93 0.186 Block*Pretreatment 2 398.38 199.19 15.67 0.000 Stain 3 266.01 88.67 6.98 0.006 Pretreatment*Stain 3 62.79 20.93 1.65 0.231 Error 12 152.52 12.71 Total 23 2038.72 Diploma in Statistics Design and Analysis of Experiments

  16. Extending the treatment structure 4 Stain levels ↔ two 2-level factors: Stain type 1 or 2 number of Coats applied 1 or 2 Diploma in Statistics Design and Analysis of Experiments

  17. Expanded Unit and Treatment Structure Units Blocks Boards Panels Factor Pretreatment Stain x Coat ANOVA MS(Blocks) MS(Pretreatment) MS(B x PT) MS(Stain) MS(Coat) MS(Interactions) MS(Panels) Diploma in Statistics Design and Analysis of Experiments

  18. Minitab model Block Pretreatment Block*Pretreatment Stain Coat Stain*Coat Pretreatment*Stain Pretreatment*Coat Pretreatment*Stain*Coat Diploma in Statistics Design and Analysis of Experiments

  19. Analysis of Variance Source DF SS MS F P Block 2 376.99 188.49 0.95 0.514 Pretreatment 1 782.04 782.04 3.93 0.186 Block*Pretreatment 2 398.38 199.19 15.67 0.000 Stain 1 38.00 38.00 2.99 0.109 Coat 1 214.80 214.80 16.90 0.001 Stain*Coat 1 13.20 13.20 1.04 0.328 Pretreatment*Stain 1 43.20 43.20 3.40 0.090 Pretreatment*Coat 1 18.38 18.38 1.45 0.252 Pretreatment*Stain*Coat 1 1.21 1.21 0.10 0.762 Error 12 152.52 12.71 Total 23 2038.72 Diploma in Statistics Design and Analysis of Experiments

  20. Laboratory 2, Exercise 1:Soup mix packet filling machine Questions: What factors affect soup powder fill variation? How can fill variation be minimised? Potential factors A: Number of ports for adding oil, 1 or 3, B: Mixer vessel temperature, ambient or cooled, C: Mixing time, 60 or 80 seconds, D: Batch weight, 1500 or 2000 lbs, E: Delay between mixing and packaging, 1 or 7 days. Response: Spread of weights of 5 sample packets Diploma in Statistics Design and Analysis of Experiments

  21. Minitab analysis Diploma in Statistics Design and Analysis of Experiments

  22. Minitab analysis Normal plot vs Pareto Principle vs Lenth? Diploma in Statistics Design and Analysis of Experiments

  23. Minitab analysis Estimated Effects for Y Term Effect Alias E -0.470 E + A*B*C*D B*E 0.405 B*E + A*C*D D*E -0.315 D*E + A*B*C Diploma in Statistics Design and Analysis of Experiments

  24. Graphical and numerical summaries Diploma in Statistics Design and Analysis of Experiments

  25. Best conditions Best conditions: B Low, D High, E High. Best conditions with E Low: B High, D Low. Diploma in Statistics Design and Analysis of Experiments

  26. Reduced model Fit model using active terms: B + D + E + BE + DE DE confirmed as active. Diploma in Statistics Design and Analysis of Experiments

  27. Diagnostics Diploma in Statistics Design and Analysis of Experiments

  28. Diagnostics Diploma in Statistics Design and Analysis of Experiments

  29. Delete Design point 5, iterate analysis • Effect estimates similar • Interaction patterns similar • s = 0.15, df = 9 ( = 14 – 5 ) Least Squares Means for Y Mean SE Mean B*D*E - - - 1.7000 0.1532 + - - 1.2050 0.1083 - + - 1.9750 0.1083 + + - 1.2250 0.1083 - - + 0.9750 0.1083 + - + 1.3600 0.1083 - + + 0.6900 0.1083 + + + 0.9400 0.1083 1.205  2.26×0.15/√2 = 0.965 to 1.445 0.69  2.26×0.15/√2 = 0.37 to 1.01 Diploma in Statistics Design and Analysis of Experiments

  30. Laboratory 2, Exercise 2Cambridge Grassland Experiment 3 grassland treatments Rejuvenator R Harrow H no treatment C randomly allocated to 3 neighbouring plots, replicated in 6 neighbouring blocks 4 fertilisers Farmyard manure F Straw S Artificial fertiliser A no fertiliser C randomly allocated to 4 sub plots within each plot. Diploma in Statistics Design and Analysis of Experiments

  31. Cambridge Grassland Experiment Diploma in Statistics Design and Analysis of Experiments

  32. Randomised Blocks analysis forTreatments Source DF SS MS F P WB 5 149700 29940 15.99 0.000 WT 2 49884 24942 13.32 0.002 WB*WT 10 18725 1872 ** Error 0 * * Total 17 218309 Model: WB + WT + WBxWT Diploma in Statistics Design and Analysis of Experiments

  33. Diagnostics Diploma in Statistics Design and Analysis of Experiments

  34. Diagnostics Diploma in Statistics Design and Analysis of Experiments

  35. WB x WT Interaction Plot Diploma in Statistics Design and Analysis of Experiments

  36. WT x WB Interaction Plot Diploma in Statistics Design and Analysis of Experiments

  37. Split Plots Analysis Model: B + T + B x T + F + B x F + T x F Source DF SS MS F P B 5 37425.1 7485.0 21.37 0.002 x T 2 12471.0 6235.5 13.32 0.002 B*T 10 4681.1 468.1 1.94 0.079 F 3 56022.7 18674.2 151.24 0.000 B*F 15 1852.1 123.5 0.51 0.914 T*F 6 781.5 130.3 0.54 0.774 Error 30 7239.6 241.3 Total 71 120473.3 Diploma in Statistics Design and Analysis of Experiments

  38. Diagnostics Diploma in Statistics Design and Analysis of Experiments

  39. Same diagnostic, Different interpretation? Diploma in Statistics Design and Analysis of Experiments

  40. Treatment comparisons • First step: summary statistics Variable Treatment Count Mean Y C 24 181.46 H 24 150.96 R 24 157.17 s2 = MS(B*T) = 468.1; df = 10 Diploma in Statistics Design and Analysis of Experiments

  41. Compare Harrow with Control Diploma in Statistics Design and Analysis of Experiments

  42. Compare Harrow with Control Diploma in Statistics Design and Analysis of Experiments

  43. Compare Harrow with Control Diploma in Statistics Design and Analysis of Experiments

  44. Compare Harrow with Control Diploma in Statistics Design and Analysis of Experiments

  45. Compare Harrow with Control Diploma in Statistics Design and Analysis of Experiments

  46. Laboratory 2, Exercise 3Fertiliser experiment Fertilisers potentially affecting bean yield Low High Dung (D): none 10 tons per acre Nitrochalk (N): none 0.4 cwt per acre SuperPhosphate (P): none 0.6 cwt per acre Muriate of Potash (K): none 1.0 cwt per acre Questions: What factors affect bean yield? How can bean yield be maximised? Diploma in Statistics Design and Analysis of Experiments

  47. Exercise 2, Minitab analysis Nis marginally significant. Pareto Principle suggests adding NPKand DP. Diploma in Statistics Design and Analysis of Experiments

  48. -0.75 Term Effect Coef Constant 47.250 Block -0.375 D -0.750 -0.375 N -8.000 -4.000 P 0.250 0.125 K -2.250 -1.125 D*N 1.250 0.625 D*P 4.500 2.250 D*K 0.000 0.000 N*P 2.250 1.125 N*K -1.750 -0.875 P*K 0.500 0.250 D*N*P -2.000 -1.000 D*N*K 2.000 1.000 D*P*K 2.250 1.125 N*P*K -5.500 -2.750 Diploma in Statistics Design and Analysis of Experiments

  49. Reduced model Fit model using active terms: D, N, P, K, DP, NP, NK, PK, NPK Active effects confirmed. Diagnostics unremarkable Diploma in Statistics Design and Analysis of Experiments

  50. 3-factor interaction Adding N reduces yield overall, but has a small positive effect at high P and low K. At low N, the P effect is negative at low K and positive at high K. At high N, this interaction is reversed. The best combination is no fertiliser. Diploma in Statistics Design and Analysis of Experiments

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