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Statistical Analysis

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 5 Complete Factorial Experiments, Completely Randomized Designs, Main Effects & Interactions. Complete Factorial Experiments. All combinations of the factor levels appear in the design at least once.

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Statistical Analysis

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  1. Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 5 Complete Factorial Experiments, Completely Randomized Designs, Main Effects & Interactions

  2. Complete Factorial Experiments All combinations of the factor levels appear in the design at least once • Does not require that each combination occur an equal number of times • Does not state what type of statistical design is being used “Factorial Design” is not a correct statistical term

  3. Completely Randomized Design • Can be any type of complete or fractional factorial experiment Randomize the sequence of test runs, assignment to experimental units

  4. Randomization Inexpensive Insurance • Bias due to changes in uncontrollable Factors or in experimental conditions • Bias due to premature termination of the experiment • Bias due to machine drift, fatigue, wear Validates key assumptions (Independence, Randomization Distributions)

  5. Changes in Experimental Conditions : Oil Viscosity Tests Experiment : Two Oils One Test Run Per Day Six Test Runs Per Week Day Week #1 Week #2 1 56.4 60.6 2 44.7 54.5 3 67.5 62.8 4 76.6 65.2 5 71.3 78.6 6 72.8 80.3 Average 64.9 67.0 (A) (A) (A) (A) (A) (A) (B) (B) (B) (B) (B) (B) t-Test: averages are not significantly different

  6. Changes in Experimental Conditions : Oil Viscosity Tests Environmental or equipment change between weeks : add 5 unit bias to all measurements in Week #2 Day Week #1 Week #2 1 56.4 65.6 2 44.7 59.5 3 67.5 67.8 4 76.6 70.2 5 71.3 83.6 6 72.8 85.3 Average 64.9 72.0 (A) (A) (A) (A) (A) (A) (B) (B) (B) (B) (B) (B) Average viscosity for Oil B is significantly greater than for Oil A

  7. Changes in Experimental Conditions : Oil Viscosity Tests Randomize the test sequence, 3 oils each week, add 5 unit bias to all measurements in Week #2 Day Week #1 Week #2 1 80.3 67.8 2 67.5 77.8 3 56.4 70.2 4 54.5 76.3 5 76.6 65.6 6 78.6 49.7 Average 67.4 69.5 (B) (A) (A) (B) (A) (B) (B) (A) (B) (A) (B) (A) Original Biased Randomized Oil B - Oil A 2.1 7.1 2.1

  8. Premature Termination :Oil Viscosity Tests Oil Type Lubricant Viscosity A #1 54.6 54.3 #2 44.7 47.0 #3 67.5 67.2 #4 76.6 77.4 B #1 60.6 #2 #3 #4 Equipment Failure

  9. Response Drift 50 Gallon Drum of Chemicals Background Noise Time Test Runs Figure 4.7 Influence of machine drift; test runs indicated by arrows.

  10. Rotating Shaft Stationary Sleeve Torque Study Goal : Investigate the effects of three factors on torque forces on rotating shafts Lay Out the Design Factor Levels Shaft Alloy Steel, Aluminum Sleeve Metal Porous, Nonporous Lubricant Type Lub 1, Lub 2, Lub 3, Lub 4 MGH Figure 5.1

  11. Construction of Completely Randomized Designs • List the factor-level combinations • All combinations, if complete factorial; only those to be tested, if a fractional factorial • Include repeat tests, if any • Number the combinations (including repeats) from 1 to N • Obtain one or more random number sequences of numbers from 1 to N • Randomize the test run sequence, if testing is performed sequentially (one after another) • Randomize the assignment of factor-level combinations to experimental units, if any

  12. Completely Randomized Design for Torque Study MGH Table 5.1 Random Number Sequence : 8, 13, 4, 7, 5, 1, 11, 15 9, 3, 12, 10, 6, 14, 16, 2

  13. Completely Randomized Design for Torque Study Repeats : Same Procedure MGH Table 5.2

  14. Completely Randomized Design for Torque Study Randomly Selected Repeat Tests cf. Table 5.3 CRD with 2 Repeats

  15. Lubricant Deposit Study What are the Main Factor Effects ?

  16. Notation(One-Factor Experiment) Factor Level 1 2 3 k Overall Average . . . . . . Average i = Factor Level Data: yij j = Repeat

  17. Factor Main Effects Change in the average response due to changes in levels of one factor Factor Level 1 2 ... k Mean m1m2 ... mk Average ... Are the means different ? Are the averages significantly different ? (Main Effects)

  18. Factor Main Effects Fixed Effects : Constant (mean) changes (Pre-selected levels, systematic changes) Random Effects : Random changes (Standard Deviation > 0) (Effects sampled from a probability distribution)

  19. Factor Main Effects Models One factor model Assumption E(eij) = 0 Conventions mi = m + ai Sai = 0 Cell Means Model Effects Model Averages

  20. Main Effects Note: mi = mj ai = aj Factor Main Effects Models Note: mi = mai = 0 Main Effects Theoretical: changes in factor-level means Empirical : changes in factor-level averages

  21. Pilot Plant Experiment 45 80 C2 Catalyst 52 83 54 68 C1 40 Concentration 60 72 20 MGH Figure 5.3 160 180 Temperature

  22. Pilot Plant Experiment 45 80 C2 Catalyst 52 83 54 68 C1 40 Concentration 60 72 20 160 180 Temperature

  23. Pilot Plant Experiment 45 80 C2 Catalyst 52 83 54 68 C1 40 Concentration 60 72 20 160 180 Temperature

  24. Pilot Plant Experiment 45 80 C2 Catalyst 52 83 54 68 C1 40 Concentration 60 72 20 160 180 Temperature

  25. Pilot Plant ExperimentMain Effects Change in the average response due to changes in levels of one factor (High – Low) Main effects do notmeasure joint factor effects Main effects are averaged across levels of the other factors

  26. Interactions Effects of the levels of one factor on the response depend on the levels of one or more other factors

  27. No Interaction Response Factor #2 Change in average response for factor #1 is constant for all levels of factor #2 Change Factor #1 Level #2 Level #1

  28. Pilot Plant Experiment 45 80 C2 62.5 Catalyst 67.5 52 83 54 68 C1 61 Change with concentration roughly the same for each catalyst 40 66 Concentration 60 72 20 160 180 Temperature

  29. Pilot Plant Experiment 45 80 C2 81.5 48.5 Catalyst 52 83 Change with temperature greater for catalyst C2 than for catalyst C1 54 68 C1 40 57 70 Concentration 60 72 20 160 180 Temperature

  30. No Interaction Catalyst C2 Pilot Plant ExperimentInteraction Plot 75 70 Average Yield (%) 65 60 Catalyst C1 55 50 20 40 Concentration (%) MGH Figure 5.4

  31. Catalyst C2 Interaction Pilot Plant ExperimentInteraction Plot 80 75 70 Average Yield (%) 65 Catalyst C1 60 55 50 160 180 Temperature (deg C) MGH Figure 5.4

  32. No Interaction Tool Type B } 15 Hours Cutting Tool Life 50 40 30 Tool Life (hrs) 20 10 Tool Type A 0 500 600 700 800 900 1000 Lathe Speed (rpm) MGH Figure 5.2

  33. Plasticity Study Purpose : Study “Plastic-Like” Properties of Friction-Reducing Lubricants and Additives Moveable Test Sample Stationary Platform Lubricant + Additive Responses: Plastic Viscosity Gel

  34. Plasticity Experiment • Design Factors • Lubricant • Acme XLT, Monarch1, Standard • Additive • None, 1%, 5% • Design • Factorial Experiment • 2 Repeat Tests / Combination • Completely Randomized Design

  35. Effects of Additives 1% - None : 4.04 - 5.39 = -1.35 5% - None : 6.39 - 5.39 = 1.00 Main Effects for Additives Factor Effects on Plastic Viscosity Effects of Lubricants Main Effects for Lubricants Acme XLT - Standard : 6.22 - 4.81 = 1.41 Monarch 1 - Standard : 4.79 - 4.81 = -0.02

  36. Factor Effects on Plastic Viscosity Primary Conclusion Acme XLT has a greater mean plastic viscosity, especially with 5% concentration If Verified by statistical analysis Weak, if any, interaction effects 8 Acme XLT 6 Average Plastic Viscosity Monarch1 Standard 4 2 0 None 1 % 5 % Additive Amount

  37. Effects of Additives 1% - None : 8.49 - 15.70 = -7.21 5% - None : 7.37 - 15.70 = -8.33 Main Effects for Additives Factor Effects on Gel Effects of Lubricants Main Effects for Lubricants Acme XLT - Standard : 11.45 - 10.01 = 1.44 Monarch 1 - Standard : 10.10 - 10.01 = 0.09

  38. Factor Effects on Gel Primary Conclusion Acme XLT has a lower mean gel with no additive, greater with additives If verified by statistical analysis Strong Interaction Effects 20 15 Average Gel Acme XLT 10 Monarch1 5 Standard 0 None 1 % 5 % Additive Amount

  39. Completely Randomized Design for Torque Study

  40. Interaction Effects :Torque Study Aluminum Shaft 85 Porous Sleeve 80 Average Torque (in-oz) 75 70 65 60 1 2 3 4 Lubricant Type

  41. Interaction Effects :Torque Study Aluminum Shaft 85 Porous Sleeve 80 Average Torque (in-oz) 75 70 65 Nonporous Sleeve 60 1 2 3 4 Lubricant Type

  42. Interaction Effects :Torque Study Average Torque (in-oz) Average Torque (in-oz) Aluminum Shaft Steel Shaft 85 85 Porous Sleeve Porous Sleeve 80 80 75 75 70 70 65 Nonporous Sleeve 65 Nonporous Sleeve 60 60 1 2 3 4 1 2 3 4 Lubricant Type Lubricant Type

  43. Interaction Effects Interaction effects cannot be properly evaluated if the design does not permit their estimation Complete factorials permit the evaluation of all main effects and all interaction effects

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