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Pertemuan 11 Analisis Varians Data Nonparametrik

Pertemuan 11 Analisis Varians Data Nonparametrik. Matakuliah : A0392 – Statistik Ekonomi Tahun : 2006. Outline Materi : Uji Kruskal Wallis Pembuatan peringkat data Statistik uji Kruskal Wallis. Analysis of Variance:Data Nonparametric. (continued).

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Pertemuan 11 Analisis Varians Data Nonparametrik

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

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

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

  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. 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 Tj2

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

  7. Kruskal-Wallis Rank Test Procedure (continued) • Critical Value for a Given a • Upper tail • Decision Rule • Reject H0: M1 = M2 = ••• = Mc if test statistic H > • Otherwise, do not reject H0

  8. 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? Kruskal-Wallis Rank Test: Example Machine1Machine2Machine3 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

  9. Example Solution: Step 1 Obtaining a Ranking Raw Data Ranks Machine1Machine2Machine3 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 Machine1Machine2Machine3 14 9 2 15 6 7 12 10 1 11 8 4 13 5 3 65 38 17

  10. Example Solution: Step 2 Test Statistic Computation

  11. H0: M1 = M2 = M3 H1: Not all equal  = .05 df = c - 1 = 3 - 1 = 2 Critical Value(s): Kruskal-Wallis Test Example Solution Test Statistic: H = 11.58 Decision:  = .05. Reject at Conclusion:  = .05 There is evidence that population medians are not all equal.

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

  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. 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. 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.j2 is the square of the rank total for group j • FR can be approximated by a chi-square distribution with (c–1) degrees of freedom • The rejection region is in the right tail

  16. 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? Friedman Rank Test: Example Machine1Machine2Machine3 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. Friedman Rank Test: Computation Table

  18. H0: M1 = M2 = M3 H1: Not all equal  = .05 df = c - 1 = 3 - 1 = 2 Critical Value: Friedman Rank Test Example Solution Test Statistic: FR= 8.4 Decision:  = .05 Reject at Conclusion:  = .05 There is evidence that population medians are not all equal.

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

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