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

Lubricant Deposit Study

Lubricant Average Deposit Ratings 50 Average Deposit Rating Lubricant Which lubricant averages are significantly different from one another ? Which lubricant averages are significantly different from one another ?

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

Experimentwise Error Rate : k Independent Statistical Tests

Assumes independence

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

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

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

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

Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 # LSD = 1.998{766.19(2/9)} 1/2 = 26.07

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

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

Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 # Based on Fisher’s least significant difference procedure, lubricant #1 produced an average deposit measurement of 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 to for lubricants 2 – 6, are not significantly different from one another.

Least Significant Interval Plot 50 Average Deposit Rating Lubricant LSD/2

Least Significant Interval Plot 50 Average Deposit Rating Lubricant LSD/2

Studentized Range Statistic Assume Studentized Range unequal n i

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

Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 # TSD = 4.441{766.19/9} 1/2 = 40.98

Lubricant Comparisons Table of Averages #1 #8 #7 #3 #4 #2 #6 # Not significantly different TSD = 4.441{766.19/9} 1/2 = 40.98

Bonferroni Method

Bonferroni Multiple Comparisons (BSD) Number of Pairwise Comparisons

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

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

Pilot Plant Chemical-Yield Study MGH Table 6.4

Main Effects Plot Average Yield C1C1 C2C2 TemperatureConcentrationCatalyst M(Temp) = 23.0 M(Conc) = -5.0 M(Cat) = 1.5

Pilot Plant Chemical-Yield Study Concentration : 20% 40% Average Yield :66.75%61.75% Average Yield (%) Temperature ( o F) Catalyst 1 Catalyst 2

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

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

Location Differences ? Interaction ?

Analysis of Variance Table for Weld Strength Experiment MGH Table 6.10