1. 1 Pertemuan 11 Analisis Varians Data Nonparametrik Matakuliah: A0392 – Statistik Ekonomi Tahun: 2006

2. 2 Outline Materi :  Uji Kruskal Wallis  Pembuatan peringkat data  Statistik uji Kruskal Wallis

3. 3 Analysis of Variance:Data Nonparametric Kruskal-Wallis Rank Test for Differences in c Medians Friedman Rank Test for Differences in c Medians (continued)

4. 4 Kruskal-Wallis Rank Test Assumptions –Independent random samples are drawn –Continuous dependent variable –Data may be ranked both within and among samples –Populations have same variability –Populations have same shape Robust with Regard to Last 2 Conditions –Use F test in completely randomized designs and when the more stringent assumptions hold

5. 5 Kruskal-Wallis Rank Test Procedure Obtain Ranks –In event of tie, each of the tied values gets their average rank Add the Ranks for Data from Each of the c Groups –Square to obtain T j 2

6. 6 Kruskal-Wallis Rank Test Procedure Compute Test Statistic – – # of observation in j –th sample –H may be approximated by chi-square distribution with df = c –1 when each n j >5 (continued)

7. 7 Kruskal-Wallis Rank Test Procedure Critical Value for a Given  – Upper tail Decision Rule –Reject H 0 : M 1 = M 2 = = M c if test statistic H > –Otherwise, do not reject H 0 (continued)

8. 8 Machine1 Machine2 Machine3 25.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40 Kruskal-Wallis Rank Test: Example As production manager, you want to see if 3 filling machines have different median filling times. You assign 15 similarly trained & experienced workers, 5 per machine, to the machines. At the.05 significance level, is there a difference in median filling times?

9. 9 Machine1 Machine2 Machine3 14 9 2 15 6 7 12 10 1 11 8 4 13 5 3 Example Solution: Step 1 Obtaining a Ranking Raw DataRanks 6538 17 Machine1 Machine2 Machine3 25.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40

10. 10 Example Solution: Step 2 Test Statistic Computation

11. 11 Kruskal-Wallis Test Example Solution H 0 : M 1 = M 2 = M 3 H 1 : Not all equal  =.05 df = c - 1 = 3 - 1 = 2 Critical Value(s): Reject at Test Statistic: Decision: Conclusion: There is evidence that population medians are not all equal.  =.05  =.05. H = 11.58

12. 12 Kruskal-Wallis Test in PHStat PHStat | c-Sample Tests | Kruskal-Wallis Rank Sum Test … Example Solution in Excel Spreadsheet

13. 13 Friedman Rank Test for Differences in c Medians Tests the equality of more than 2 (c) population medians Distribution-Free Test Procedure Used to Analyze Randomized Block Experimental Designs Use  2 Distribution to Approximate if the Number of Blocks r > 5 – df = c – 1

14. 14 Friedman Rank Test Assumptions –The r blocks are independent –The random variable is continuous –The data constitute at least an ordinal scale of measurement –No interaction between the r blocks and the c treatment levels –The c populations have the same variability –The c populations have the same shape

15. 15 Friedman Rank Test: Procedure  Replace the c observations by their ranks in each of the r blocks; assign average rank for ties  Test statistic:  R.j 2 is the square of the rank total for group j  F R can be approximated by a chi-square distribution with (c – 1) degrees of freedom  The rejection region is in the right tail

16. 16 Friedman Rank Test: Example As production manager, you want to see if 3 filling machines have different median filling times. You assign 15 workers with varied experience into 5 groups of 3 based on similarity of their experience, and assigned each group of 3 workers with similar experience to the machines. At the.05 significance level, is there a difference in median filling times? Machine1 Machine2 Machine3 25.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40

17. 17 Friedman Rank Test: Computation Table

18. 18 Friedman Rank Test Example Solution H 0 : M 1 = M 2 = M 3 H 1 : Not all equal  =.05 df = c - 1 = 3 - 1 = 2 Critical Value: Reject at Test Statistic: Decision: Conclusion: There is evidence that population medians are not all equal.  =.05 F R = 8.4

19. 19 Chapter Summary Discussed Kruskal-Wallis Rank Test for Differences in c Medians Illustrated Friedman Rank Test for Differences in c Medians (continued)