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Statistical Analysis

Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 11 Multiple Comparisons & Class Exercises. Lubricant Deposit Study. Lubricant Deposit Study. Lubricant Average Deposit Ratings. 100. Which lubricant averages are significantly different

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Statistical Analysis

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  1. Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 11 Multiple Comparisons & Class Exercises

  2. Lubricant Deposit Study

  3. Lubricant Deposit Study

  4. Lubricant Average Deposit Ratings 100 Which lubricant averages are significantly different from one another ? 75 Average Deposit Rating 50 25 1 2 3 4 5 6 7 8 Lubricant

  5. Multiple Comparisons Several comparisons of factor means or of factor effects using procedures that control the overall significance or confidence level Comparisonwise Error Rate aC = Pr(Type 1 Error) for One Statistical Test Experimentwise Error Rate aE = Pr(One or More Type 1 Errors) for Two or More Tests

  6. Experimentwise Error Rate :k Independent Statistical Tests

  7. Experimentwise Error Rate :k Independent Statistical Tests

  8. Experimentwise Error Rate :k Independent Statistical Tests Assumes independence

  9. Many Comparisons • Overall Type I Error Rate (experimentwise error rate, aE) for k tests is much greater than the individual test error rate (aC) • Post-Hoc (after significant F tests) comparisons are usually based on order statistics aC = .05 Assumes independence

  10. Experimentwise Error Rate :k Independent Statistical Tests Experimentwise & comparisonwise error rates Assumes independence aC = .05 Dependent Tests Common MSE Lack of orthogonality

  11. Fisher’s Least Significant Difference (LSD) Protected: Preceded by an F Test for overall significance Unprotected:Not preceded by an F Test – Individual t tests MGH Exhibit 6.9

  12. Least Significant Interval (LSI) Plot LSI Plot Plot the averages, with bars extending LSD/2 above & below each average. Bars that do NOT overlap indicate sSignificantly different averages. If Unequal ni : Use MGH Exhibit 6.13

  13. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.89 51.50 61.94 64.94 72.50 78.11 78.94 88.61 LSD = 1.998{766.19(2/9)}1/2 = 26.07

  14. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.89 51.50 61.94 64.94 72.50 78.11 78.94 88.61 Not significantly different LSD = 1.998{766.19(2/9)}1/2 = 26.07

  15. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.89 51.50 61.94 64.94 72.50 78.11 78.94 88.61 Not significantly different LSD = 1.998{766.19(2/9)}1/2 = 26.07

  16. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.89 51.50 61.94 64.94 72.50 78.11 78.94 88.61 Based on Fisher’s least significant difference procedure, lubricant #1 produced an average deposit measurement of 24.89 that is significantly less than the averages of all the other lubricants. Lubricant 8 has the second lowest average deposit (51.50), but it is not significantly different from the averages for lubricants 3, 4, and 7. The third smallest average deposit (61.94) was obtained by lubricant 7, but it is not significantly different from the averages for lubricants 2-4, 6, and 8. The five lubricants with the highest averages, ranging from 64.94 to 88.61 for lubricants 2 – 6, are not significantly different from one another.

  17. Least Significant Interval Plot 100 LSD/2 75 LSD/2 Average Deposit Rating 50 25 1 2 3 4 5 6 7 8 Lubricant

  18. Least Significant Interval Plot 100 LSD/2 75 LSD/2 Average Deposit Rating 50 25 1 2 3 4 5 6 7 8 Lubricant

  19. Studentized Range Statistic Assume Studentized Range unequal ni

  20. Tukey’s “Honest” Significant Difference (HSD or TSD) MGH Exhibit 6.11

  21. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.89 51.50 61.94 64.94 72.50 78.11 78.94 88.61 TSD = 4.441{766.19/9}1/2 = 40.98

  22. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.89 51.50 61.94 64.94 72.50 78.11 78.94 88.61 Not significantly different TSD = 4.441{766.19/9}1/2 = 40.98

  23. Bonferroni Method

  24. Bonferroni Multiple Comparisons (BSD) Number of Pairwise Comparisons

  25. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.89 51.50 61.94 64.94 72.50 78.11 78.94 88.61 BSD = 3.26{766.19(2/9)}1/2 = 42.54 m = 28: a= .00089

  26. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.89 51.50 61.94 64.94 72.50 78.11 78.94 88.61 Not significantly different BSD = 3.26{766.19(2/9)}1/2 = 42.54 a= .00089

  27. Pilot Plant Chemical-Yield Study MGH Table 6.4

  28. Main Effects Plot 80 M(Temp) = 23.0 70 Average Yield M(Conc) = -5.0 M(Cat) = 1.5 60 50 160 180 20 40 C1 C2 Temperature Concentration Catalyst

  29. Pilot Plant Chemical-Yield Study Concentration : 20% 40% Average Yield : 66.75% 61.75% 100 Average Yield (%) Catalyst 2 80 Catalyst 1 60 40 160 180 Temperature (oF)

  30. Pilot Plant Chemical-Yield Study Concentration : 20% 40% Average Yield : 66.75% 61.75% 100 Average Yield (%) Catalyst 2 80 Catalyst 1 60 Note: LSI Bars Not Necessary if All Averages are (or are not) Significantly Different 40 160 180 Temperature (oF)

  31. Weld Strength Experiment • Electrode Position • 1, 2A, 2B, 3, 4, 5, 6 • Electrode Polarity • Positive, Negative • Grid Wire Type • Coded 1, 2, 3, 4, 5 Response Weld Strength (lbs) Complete Factorial Experiment Completely Randomized Design k = 7 x 2 x 5 = 70 Combinations r = 2 Repeats n = 140 Test Runs Factors

  32. Location Differences ? Interaction ?

  33. Analysis of Variance Table for Weld Strength Experiment MGH Table 6.10

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