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

2. Lubricant Deposit Study

4. Lubricant Average Deposit Ratings 50 Average Deposit Rating Lubricant 75 100 25 12345678 Which lubricant averages are significantly different from one another ? Which lubricant averages are significantly different from one another ?

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

6. Experimentwise Error Rate : k Independent Statistical Tests

8. Assumes independence

9. Many Comparisons Overall Type I Error Rate (experimentwise error rate,  E ) for k tests is much greater than the individual test error rate (  C ) Post-Hoc (after significant F tests) comparisons are usually based on order statistics  C  Assumes independence

10. Experimentwise Error Rate : k Independent Statistical Tests Experimentwise & comparisonwise error rates Dependent Tests Common MS E Lack of orthogonality Common MS E Lack of orthogonality  C  Assumes independence

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 n i : Use MGH Exhibit 6.13

13. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.8951.5061.9464.9472.5078.1178.9488.61 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 24.8951.5061.9464.9472.5078.1178.9488.61 LSD = 1.998{766.19(2/9)} 1/2 = 26.07 Not significantly different

15. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.8951.5061.9464.9472.5078.1178.9488.61 LSD = 1.998{766.19(2/9)} 1/2 = 26.07 Not significantly different

16. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.8951.5061.9464.9472.5078.1178.9488.61 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 50 Average Deposit Rating Lubricant 75 100 25 12345678 LSD/2

18. Least Significant Interval Plot 50 Average Deposit Rating Lubricant 75 100 25 12345678 LSD/2

19. Studentized Range Statistic Assume Studentized Range unequal n i

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 24.8951.5061.9464.9472.5078.1178.9488.61 TSD = 4.441{766.19/9} 1/2 = 40.98

22. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.8951.5061.9464.9472.5078.1178.9488.61 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 24.8951.5061.9464.9472.5078.1178.9488.61 BSD = 3.26{766.19(2/9)} 1/2 = 42.54 m  =.00089

26. Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 #5 24.8951.5061.9464.9472.5078.1178.9488.61 Not significantly different BSD = 3.26{766.19(2/9)} 1/2 = 42.54  =.00089

29. Pilot Plant Chemical-Yield Study MGH Table 6.4

30. Main Effects Plot Average Yield 50 60 70 80 1601802040C1C1 C2C2 TemperatureConcentrationCatalyst M(Temp) = 23.0 M(Conc) = -5.0 M(Cat) = 1.5

31. Pilot Plant Chemical-Yield Study Concentration : 20% 40% Average Yield :66.75%61.75% 160180 60 Average Yield (%) Temperature ( o F) 80 100 40 Catalyst 1 Catalyst 2

32. Pilot Plant Chemical-Yield Study Concentration : 20% 40% Average Yield :66.75%61.75% 160180 60 Average Yield (%) Temperature ( o F) 80 100 40 Catalyst 1 Catalyst 2 Note: LSI Bars Not Necessary if All Averages are (or are not) Significantly Different

34. 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) Factors Complete Factorial Experiment Completely Randomized Design k = 7 x 2 x 5 = 70 Combinations r = 2 Repeats n = 140 Test Runs Complete Factorial Experiment Completely Randomized Design k = 7 x 2 x 5 = 70 Combinations r = 2 Repeats n = 140 Test Runs

35. Location Differences ? Interaction ?

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