1. 1 1 Slide MULTIPLE COMPARISONS

2. 2 2 Slide Multiple Comparison Procedures n nSuppose that analysis of variance has provided statistical evidence to reject the null hypothesis of equal population means. n nFisher’s least significant difference (LSD) procedure can be used to determine where the differences occur.

3. 3 3 Slide Fisher’s LSD Procedure nTest Statistic nHypotheses =

4. 4 4 Slide Fisher’s LSD Procedure where the value of t a /2 is based on a t distribution with n T - k degrees of freedom. nRejection Rule Reject H 0 if p -value <  p -value Approach: Critical Value Approach: Reject H 0 if t t a /2

5. 5 5 Slide nTest Statistic Fisher’s LSD Procedure Based on the Test Statistic x i - x j __ where Reject H 0 if > LSD nHypotheses nRejection Rule =

6. 6 6 Slide Fisher’s LSD Procedure Based on the Test Statistic x i - x j Reed Manufacturing Recall that Janet Reed wants to know if there is any significant difference in the mean number of hours worked per week for the department managers at her three manufacturing plants. Analysis of variance has provided statistical evidence to reject the null hypothesis of equal population means. Fisher’s least significant difference (LSD) procedure can be used to determine where the differences occur.

7. 7 7 Slide For  =.05 and n T - k = 15 – 3 = 12 degrees of freedom, t. 025 = 2.179 MSE value was computed earlier Fisher’s LSD Procedure Based on the Test Statistic x i - x j

8. 8 8 Slide nLSD for Plants 1 and 2 Fisher’s LSD Procedure Based on the Test Statistic x i - x j Conclusion Conclusion Test Statistic Test Statistic = |55  68| = 13 Reject H 0 if > 6.98 Rejection Rule Rejection Rule Hypotheses (A) Hypotheses (A) The mean number of hours worked at Plant 1 is The mean number of hours worked at Plant 1 is not equal to the mean number worked at Plant 2. =

9. 9 9 Slide nLSD for Plants 1 and 3 Fisher’s LSD Procedure Based on the Test Statistic x i - x j Conclusion Conclusion Test Statistic Test Statistic = |55  57| = 2 Reject H 0 if > 6.98 Rejection Rule Rejection Rule Hypotheses (B) Hypotheses (B) There is no significant difference between the mean number of hours worked at Plant 1 and the mean number of hours worked at Plant 3. =

10. 10 Slide nLSD for Plants 2 and 3 Fisher’s LSD Procedure Based on the Test Statistic x i - x j Conclusion Conclusion Test Statistic Test Statistic = |68  57| = 11 Reject H 0 if > 6.98 Rejection Rule Rejection Rule Hypotheses (C) Hypotheses (C) The mean number of hours worked at Plant 2 is not equal to the mean number worked at Plant 3. =

11. 11 Slide MULTIPLE COMPARISONS PRACTICE

12. 12 Slide Perform Multiple Comparisons Analysis of Variance SourceSSd.f.MSFp-value Treatment225.682112.8422.150.000014 Error91.68185.09 Total317.3620 DirectIndirectCombination 1720.425 s2s2 5.016.264.01 n777  =0.05

13. 13 Slide F Distribution

14. 14 Slide TYPE I ERRORS

15. 15 Slide n nThe experiment-wise Type I error rate gets larger for problems with more populations (larger k ). Type I Error Rates  EW = 1 – (1 –  ) ( k – 1)! The comparison-wise Type I error rate  indicates the level of significance associated with a single pairwise comparison. The experiment-wise Type I error rate  EW is the probability of making a Type I error on at least one of the ( k – 1)! pairwise comparisons.

16. 16 Slide