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Statistical Analysis

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 17 Block Designs. Lawnmower Stopping Times. Complete Factorial Experiment, Repeat Tests. MGH Table 9.1. Lawnmower Stopping Times. Lawnmowers are experimental units with potentially

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Statistical Analysis

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  1. Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 17 Block Designs

  2. Lawnmower Stopping Times Complete Factorial Experiment, Repeat Tests MGH Table 9.1

  3. Lawnmower Stopping Times Lawnmowers are experimental units with potentially large unit-to-unit variability

  4. Lawnmower Stopping Times Overall Variation Within manufacturer variation B B B B B B Fixed and Random Effects B B B B B B A AA A A A A A A AA A 150 175 200 225 250 275 300 Between manufacturer variation Fixed Effect

  5. Lawnmower Stopping Times Lawnmower Variation Lawnmower variation (Within Manufacturer): SSL(M) Random Effect B B B A A A 150 175 200 225 250 275 300 Between manufacturer Variation: SSM Fixed Effect

  6. Manufacturer ComparisonRepeat Test Variation Speed and Uncertainty Variation H H Random Effect L L H H L L L L HH HH L LL L L H H H 150 175 200 225 250 275 300 Cutoff Times (.01 sec) Speed variation Variation: SSS Fixed Effect

  7. Statistical Model Manufacturer Lawnmower Speed Repeat Test Random Components of Variation Lawnmowers (within manufacturers) Repeat Test Variation

  8. Combined Error Distribution Likelihood Combined Error Includes All Sources of Variation

  9. Repeat Test Error Distribution Likelihood Repeat Test Error Only Source of Uncertainty for Speed Comparisons

  10. Replicate and Repeat Test Error Distributionss Repeat Tests Likelihood Lawnmowers Error Both Contribute to the Uncertainty in Manufacturer Comparisons

  11. Combined Error Distribution Repeat Tests Likelihood Lawnmowers Combined Error eC = eL + eRsC2 = sL2 + sR2

  12. Controlling Experimental Variability • BlockingMatching Experimental Units to obtain Homogeneous Units in each block • GroupingSequencing of test runs to achieve more uniformity within each group • ReplicationRepeating of the entire experiment or portions of it under possibly dissimilar conditions • Repeat TestsTwo or more factor-level combinations repeated under “identical” conditions

  13. Block Designs Blocks: Groups of homogeneous experimental units or groups of test runs conducted under similar conditions Conditions within blocks are more similar than conditions between blocks Design Issue : Factor changes within a block only subject to repeat test variation Factor changes between blocks also subject to block variation

  14. Blocking Designs Key Assumptions Blocks are not random FACTOR effects Blocks do not interact with the design factors Blocks only contribute to variability If Blocks interact with the design factors, then the block factor should be treated as an additional design factor.

  15. Purposes • Reduce experimental variation in the comparisons of factor effects ith Factor Effect Error Variation From All Sources

  16. Purposes • Reduce experimental variation in the comparisons of factor effects ith Factor Effect jth Block Effect Error Variation From All Remaining Sources (Except Blocks) bj : Random Block Effect

  17. Purposes • Reduce experimental variation in the comparisons of factor effects by changing factor levels within each block • Obtain estimates of each source of variation • Often necessitated by experimental conditions, restrictions Blocks are Not Additional Random Experimental Factors

  18. Randomized Complete Block Designs Complete or Fractional Factorial Experiment in Each Block • Group experimental units into homogeneous blocks, if applicable • Groups test run sequence into homogeneous blocks, if applicable • Randomly assign factor-level combinations to the experimental units and/or test run sequence • Use a separate randomization for each block

  19. Allergic-Reaction Study • Develop a laboratory protocol to study allergic reactions to environmental pollutants • 10 mice from a single strain • Two locations (ear, back) on each animal • Duplicate skin thickness measurements at each location (with caliper) • Measurements immediately after injection with a skin irritant

  20. Allergic-Reaction Study • Develop a laboratory protocol to study allergic reactions to environmental pollutants • 10 mice from a single strain • Two locations (ear, back) on each animal • Duplicate skin thickness measurements at each location (with caliper) • Measurements immediately after injection with a skin irritant Factor: Location Blocks: Animals Repeats: Duplicate measurements

  21. Design Layout for Allergic-Reaction Study Replicate (Block) Repeat Tests Randomized Complete Block Design

  22. Allergic-Reaction Study

  23. Analysis General Model Specification Blocks: Main Effect, No Interactions Factors: Main Effects, & Interactions

  24. Allergic-Reaction Study

  25. Designs with Complete or Fractional Factorials • Fractional factorial experiments in completely randomized designs • Tables of designs : Appendix 7.A.1 • Aliasing among all factor effects determined by the defining equations • Complete factorial experiments in randomized incomplete block designs • Tables of designs : Appendix 9.A.1 • Only aliasing is between blocks and the defining equations • Fractional factorial experiments in randomized incomplete block designs • Tables of designs : Appendix 9.A.2 • Aliasing between blocks and the defining equations, and the usual aliasing among factor effects determined by the defining equations

  26. Drilling Tool Experiment Drill Pipe Tool Joint Drill Angle Factors Levels Rotational Drill Speed 60, 75 rpm Longitudinal Velocity 50, 100 feet/minute Drill Pipe Length 200, 400 feet Drilling Angle 30, 60 degrees Tool Joint Geometry Straight, Ellipsoidal Edges

  27. Drilling Tool Experiment Restriction No more than 20 test runs per day • Half fraction each day • Defining Contrast : I = ABCDE • ABCDE aliased with Blocks • Randomly select 4 repeat tests each day • Randomize • Half fractions to days • Test run sequence each day Design ?

  28. Drilling Tool Experiment First Day: 20 Test Runs • Half Fraction • Defining contrast : I = ABCDE • Resolution V • 4 repeat tests • Analysis • Main Effects (5 df) • Two-factor interactions (10 df) • Estimate of error standard deviation (4 df)

  29. } Combine ? Drilling Tool Experiment Complete Factorial: 40 Test Runs • Randomized incomplete block design • Day alias : ABCDE • Resolution not a meaningful property – Complete factorial in incomplete blocks, not a fractional factorial • 8 repeat tests • Analysis • Main Effects (5 df) • Day Effect = ABCDE (1 df) • Two-Factor interactions (10 df) • Three-Factor interactions (10 df) • Four-Factor interactions (5 df) • Estimate of error standard deviation (8 df)

  30. Drilling Tool Experiment :Second Scenario Restriction No More than 10 Test Runs per Day • Quarter fraction each day • Defining contrasts : I = ABD=ACE (= BCDE) • Assign factors so that ABD, ACE are of little interest (WHY ?) • Randomly select 2 repeat tests each day • Randomize • Quarter fractions to days • Test run sequence each day

  31. Randomized Incomplete Block Design ABD = -1 ACE = -1 (BCDE = +1) ABD = -1 ACE = +1 (BCDE = -1) ABD = +1 ACE = -1 (BCDE = -1) ABD = +1 ACE = +1 (BCDE = +1) Day 1 Day 2 Day 3 Day 4 Sequence : 2 4 3 1

  32. Drilling Tool Experiment First Block: 10 Test Runs • Quarter fraction • Defining contrast : I = ABD=ACE=BCDE • Resolution III • 2 repeat tests • Analysis • Main effects (5 df) • Estimate of error standard deviation • 2 df -- no assumptions • 4 df -- assuming NO interaction effects

  33. Drilling Tool Experiment Second Day: 10 more test runs • Half Fraction • Choose 2nd day so that BCDE has one sign for both days • Defining contrast : I = BCDE • Resolution IV (Half Fraction) & ABD, ACE Aliased with Days • 4 Repeat tests • Analysis (n = 20) • Main effects (5 df), Day effect = BCDE (1 df) • Two-factor interactions (9 df) : BC = DE Aliased • Estimate of error standard deviation (4 df)

  34. Drilling Tool Experiment Third Day: 10 More Test Runs • 3/4 Fraction • ABD, ACE, BCDE aliased with days • Some interactions partially aliased with one another • 6 repeat tests • Analysis (n = 30) • Main effects (5 df), Day effects (2 df) • Two-factor interactions (10 df) • Other interactions (6) • Estimate of error standard deviation (6 df)

  35. Drilling Tool Experiment Complete Factorial: 10 More Test Runs • Randomized incomplete block design • ABD, ACE, BCDE aliased with days (Complete Factorial) • 8 repeat tests • Analysis (n = 40) • Main effects (5 df) • Day effects = ABD = ACE = BCDE (3 df) • Two-factor Interactions (10 df) • Three-factor interactions (8 df): ABD, ACE aliased with days • Four-factor interactions (4 df): BCDE aliased with days • Five-Factor interaction (1 df) • Estimate of error standard deviation (8 df)

  36. Blocking and Sequential Experimentation Run select fractions in blocks, analyze each block as it is completed can continue or terminate, as warranted by the analysis

  37. Balanced Incomplete Block Designs • b blocks • f factor-level combinations • k < f experimental units per block Used when blocks contain fewer experimental units than the number of unique factor-level combinations No interactions with the design factor(s)

  38. Balanced Incomplete Block Designs • N = fr = bk • p = r(k - 1)/(f - 1) • b > f - 1 Blocks = b Factor combinations = f Units per block = k Each combination occurs r times Each pair occurs together in p blocks Note: Cannot select arbitrary values for b, f, k, r, and p

  39. Balanced Incomplete Block Designs Blocks = b Factor combinations = f Units per block = k Each combination occurs r times Each pair occurs together in p blocks • Select a basic design based on the values of b, f, and k • MGH Table 9.A.3 • Randomly order the blocks and/or the testing of blocks, as applicable • Randomly assign the combinations to the units and/or test sequence in each block

  40. Oil-Consumption Experiment • Measure the fuel consumption associated with four oils • Single test engine, single dynamometer test stand -- reduce extraneous variation • Test stand must be recalibrated after two test runs • Three replicates f = 4, k = 2, r = 3 so that b = 6, p = 1

  41. Oil-Consumption Experiment Recalibration (Block) Oils Tested 1 2 3 4 5 6 A,B C,D A,C B,D A,D B,C } Replicate 1 } Replicate 2 } Replicate 3 Randomize Replicates, Blocks within replicates, Letters within blocks, Assignment of oils to letters MGH Table 9.A.3, Design 1

  42. Latin Square Designs • Control two sources of variability • Restrictions • Factor of interest and two blocking factors each at k levels • No Interactions among the experimental and blocking factors Experiment Size Latin Square : n = k2 Complete Factorial : n = k3 + r

  43. Latin Square Design • Lay out a table with k rows and k columns • Assign the letters A, B, ... , K to the cells in the first row of the table • For the next row, move the first letter to the last position , shift all letters one position to the left • Repeat the previous step until all rows are completed • Randomize: • The levels of one blocking factor to the rows • The levels of one blocking factor to the columns • The levels of the experimental factor to the letters

  44. Latin Square Design Typical Layout First Block Levels 1 2 3 4 1 A B C D 2 3 4 Factor Levels Second Block Levels MGH Table 8A.2

  45. Latin Square Design Typical Layout First Block Levels 1 2 3 4 1 A B C D 2 B C D A 3 C D A B 4 D A B C Factor Levels Second Block Levels MGH Table 9A.4

  46. Tire-Test Study • Road test of four tire brands • One test run : several hundred miles • Several trucks needed • Several test days Latin Square Design : 4 Tire Brands 4 Trucks 4 Test Days Expected similar performance of each brand on all trucks, days (Apart from Random Variation)

  47. Tire-Test Study Day 4 3 1 2 3 Tire 4Tire 2 Tire 1 Tire 3 4Tire 2 Tire 1 Tire 3Tire 4 1 Tire 1 Tire 3Tire 4Tire 2 2 Tire 3Tire 4Tire 2 Tire 1 Tire Brands Truck

  48. Tire Effects Main Effects Only Day Effect Truck Effect Tire Effect

  49. Tire Effects Main Effects Only Day Effect Truck Effect Tire Effect Comparison of Tire 1 with Tire 2 : IF NO INTERACTIONS

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