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Statistical Analysis

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 20 Nested Designs and Analyses. Factors Shaft Alloy Sleeve Porosity Lubricant. Levels Steel, Aluminum Porous, Nonporous Lub 1, Lub 2, Lub 3, Lub 4.

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Statistical Analysis

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  1. Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 20 Nested Designs and Analyses

  2. Factors Shaft Alloy Sleeve Porosity Lubricant Levels Steel, Aluminum Porous, Nonporous Lub 1, Lub 2, Lub 3, Lub 4 Can be fixed or random effects Torque Experiment Crossed Factors Levels have the same physical meaning regardless of the other levels with which they are combined in the experiment MGH Exhibit 11.1

  3. Torque Experiment :Crossed Factors Alloy Porosity Lubricant SteelAluminum PorousNonporous 1 2 3 4

  4. Torque Experiment :Crossed Factors Alloy Porosity Lubricant SteelAluminum PorousNonporous 1 2 3 4 MGH, Fig. 11.1

  5. Factors Manufacturer Lawnmower Speed Levels A, B 1, 2, 3, 4, 5, 6 High, Low Crossed Nested Crossed Usually random effects Lawnmower Cutoff Times Experiment Nested Factors Unique levels of a factor for each level or combination of levels of one or more other factors MGH Exhibit 11.2

  6. Lawnmower Cutoff Times :Nested Factors Speed Manufacturer Lawnmower High Low AB 1 2 3 4 5 6 MGH, Fig. 11.3

  7. Factors Shift Plaque Disk 1, 2 1, 2 1, 2, 3, 4 Fixed or Random ? Crossed or Nested ? Polyethylene Density Experiment • Polyethylene pellets heated, melted, pressed into a plaque • Disks cut from plaques, density of each polyethylene disk then measured • Two shifts of technicians • Each shift prepares two plaques • 4 disks cut from each plaque Crossed Nested Nested in Shift in Plaque

  8. Polyethylene Density Experiment Shift Plaque Disk 12 1 2 3 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 MGH, Fig. 11.2

  9. Polyethylene Density Experiment • Polyethylene pellets heated, melted, pressed into a plaque • Disks cut from plaques, density of Each polyethylene disk then measured • Two shifts of technicians • Each shift prepares two plaques • 4 disks cut from each plaque Factors Shift Plaque Disk 1, 2 1, 2 1, 2, 3, 4 Fixed Random Random

  10. Hierarchically Nested Designs • Each factor is progressively nested within the levels of the preceding factor • Design is usually balanced • Equal number of levels of one factor within levels of the preceding factor • More information is obtained on factors lower in the hierarchy than those higher Source df A a - 1 B(A) a(b - 1)C(AB) ab(c - 1)

  11. Polyethylene Density Experiment Shift 12 df (Shift) = a-1 = 1 MGH, 11.2

  12. Polyethylene Density Experiment Shift Plaque 12 1 2 3 4 df (Plaques) = a(b-1) = 2(1) = 2

  13. Polyethylene Density Experiment Shift Plaque Disk 12 1 2 3 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 df (Disks) = ab(c-1) = 2 x 2 x (3) = 12

  14. Hierarchically Nested Designs • Establish the hierarchy of factors • Select, randomly if appropriate, the levels to be nested with each level of the preceding factor • Randomize the test sequence or the assignment of combinations to experimental units Each factor is progressively nested within the preceding factors

  15. Agricultural Field Trial • Investigate the yield of a new variety of crop • Factors • Insecticides • Fertilizers • Experimental Units • Farms • Fields within farms Experimental Design ? Fertilizers can be applied to individual fields; Insecticides must be applied to an entire farm from an airplane

  16. Agricultural Field Trial • Insecticides applied to farms • One-factor ANOVA • Main effect: Insecticides • MSE: Farm-to-farm variability Farms

  17. Agricultural Field Trial • Fertilizers applied to fields • One-factor ANOVA • Main Effect: Fertilizers • MSE: Field-to-field variability Fields

  18. Agricultural Field Trial • Insecticides applied to farms, fertilizers to fields • Two sources of variability • Insecticides subject to farm-to-farm variability • Fertilizers and insecticides x fertilizers subject to field-to-field variability Farms Fields

  19. Split-Plot Design Two hierarchically nested factors, with additional crossed factors occurring within levels of the nested factor An Experiment Can Have Either of these Features Two sizes of experimental units, one nested within the other, with crossed factors applied to the smaller units

  20. Split-Plot Design Whole-Plot Experiment : Whole-Plot Factor = A Level A1 Level A2 Level A2 Level A1

  21. Split Plot DesignsAnalysis of Variance Table

  22. Split-Plot Design Split-Plot Experiment : Split-Plot Factor = B B2 B1 B1 B2 B1 B1 B2 B1 B2 B2 B2 B1 B1 B2 B1 B2 Level A1 Level A2 Level A2 Level A1

  23. Split Plot DesignsAnalysis of Variance Table

  24. Agricultural Field Trial

  25. Agricultural Field Trial Insecticide 2 Insecticide 1 Insecticide 2 Insecticide 2 Insecticide 1 Insecticide 1

  26. Agricultural Field Trial Insecticide 2 Insecticide 1 Insecticide 2 Fert A Fert B Fert A Fert B Fert B Fert B Fert A Fert A Fert B Fert A Fert B Fert A Fert A Fert B Insecticide 2 Fert A Fert A Fert B Fert A Fert B Fert B Fert B Fert B Fert A Fert A Fert B Fert B Fert A Fert B Fert A Fert B Fert A Fert A Fert A Fert A Fert A Fert B Fert B Fert B Fert A Fert B Insecticide 1 Insecticide 1

  27. Agricultural Field Trial Whole Plots = Farms Large Experimental Units Split Plots = Fields Small Experimental Units

  28. Agricultural Field Trial Whole Plots = Farms Large Experimental Units Whole-Plot Factor = Insecticide Whole-Plot Error = Whole-Plot Replicates Split Plots = Fields Small Experimental Units Split-Plot Factor = Fertilizer Split-Plot Error = Split-Plot Replicates

  29. Connector Pin Experiment Factor Levels Vendor Wire 1, 2 Environment Ambient, High Humidity Heat Treatment Yes, No

  30. Connector Pin Experiment Factor Levels Vendor Wire 1, 2 Environment Ambient, High Humidity Heat Treatment Yes, No Whole Plot : Reel of Wire Split Plot : 6” Samples of Wire

  31. Connector Pin Split-Plot Design Vendor #1 Vendor #2 Vendor #2 Vendor #1 Amb, Yes Amb, Yes Amb, Yes Amb, Yes High, Yes High, Yes High, Yes High, Yes Amb, No Amb, No Amb, No Amb, No High, No High, No High, No High, No Reel #1 Reel #2 Reel #3 Reel #4 Vendor Error: Reel-to-Reel Variation Env., Heat Error: Sample-to-Sample Variation

  32. Constructing Split-Plot Designs • Identify the whole plots and the factors to be assigned to them • Randomly select a whole plot • Randomly assign a whole-plot factor combination to the whole plot • Continue until all combinations have been assigned • Identify the split plots and the factors to be assigned to them • Randomly select a split plot • Randomly assign a split-plot factor combination to the split plot • Continue until all combinations have been assigned

  33. Cylinder Wear Study Spark Plug Air-Fuel Inlet Exhaust Air-Fuel Mixture Piston Ring Piston Cylinder Wall MGH, Fig. 11.7

  34. Cylinder Wear Study Factors Levels Piston Rings (A) Type 1, Type 2 Engine Oil (B) Oil A, Oil B Engine Speed (C) 1500, 3000 rpm Intake Temperature (D) 30, 90 oF Air-Fuel Mixture (E) Lean, Rich Fuel Oxygen Content (F) 2.5, 7.5 %

  35. Cylinder Wear Study MGH Table 11.6

  36. Restricted Randomization :Split-Plot Design MGH Table 11.7

  37. Restricted Randomization • Sometimes Randomization is Restricted • Cost • Difficulty of experimentation • Inability to (or failure to) reset operating conditions, machine settings • Split plot experiments are an example • Whole plot factors must be assigned to the whole plots • Split plot factors can be randomized over any of the split plots – • Useful to have replicate or repeat runs

  38. Restricted Randomization :Split-Plot Design Whole Plot Factor Changes Split Plot Factor Changes MGH Table 11.7

  39. Mixed Models One or more factors fixed, One or more factors random Convention: Interaction of a fixed and a random factor is a random effect Two-Factor Model yijk = m + ai + bj + (ab)ij + eijk i = 1, ..., a j = 1, ..., b k = 1, ..., r

  40. Interaction both Fixed and Random (Alternative Assumption) For each i (ab)ij ~ NID(0,sab2) For each j Implications of Alternative Assumptions Interaction Random (Preferred Assumption) (ab)ij ~ NID(0,sab2) No “constraints” required Both main effects tested against the interaction effect

  41. Nested Effects Notation Enclose subscripts of the factor(s) in which an effects is nested in parentheses bj (i)ck (ij) (bg)jk (i) Nested Factor Levels Three-Factor, Hierarchically Nested Design Model yijkl = m + ai + bj(i) + ck(ij) + eijkl eijkl = el(ijk)

  42. Sums of SquaresNested Effects, Balanced Designs yijk = m + ai + bj(i) + ck(ij) + eijkl B Nested Within A Compare Averages for Nested Levels with the Overall Average for Levels having the Same Nesting

  43. Sums of SquaresNested Effects, Balanced Designs yijk = m + ai + bj(i) + eijk B Nested Within A df: a(b-1) = b-1 + (a-1)(b-1)

  44. Analysis of Variance Table

  45. Split Plot Design Model Whole Plots Design Factor: A (fixed) Whole Plot Error: B(A) (random) Split Plots Design Factor: C (fixed) yijkl = m + ai + bj(i) + gk + (ag)ik + eijkl

  46. Split Plot Design Model Whole Plots Design Factor: A (fixed) Whole Plot Error: B(A) (random) Split Plots Design Factor: C (fixed) yijkl = m + ai + bj(i) + gk + (ag)ik + eijkl Whole Plot Analysis Main Effects & Interactions of Factors Applied to the Whole Plots; Uses Whole Plot Error MS Split Plot Analysis Main Effects & Interactions of Factors Applied to the Split Plot, Interactions of Whole Plot & Split Plot Factors; Uses Error MS

  47. Automatic Cutoff Times:Split-Plot Design Factors Manufacturers: A, B Lawnmowers: 3 for Each Manufacturer Speeds: High, Low Whole Plot W.P. Error Split Plot Whole Plot Analysis Split Plot Analysis MGH Table 13.6

  48. Split-Plot ANOVA Table

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