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Wilcoxon Rank Sum Test

Wilcoxon Rank Sum Test. 1. Wilcoxon with both n 1 and n 2 < 10 2. Wilcoxon with both n 1 and n 2 ≥ 10 3. Examples. Wilcoxon Rank Sum Test. Recall from last week:

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Wilcoxon Rank Sum Test

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  1. Wilcoxon Rank Sum Test • 1. Wilcoxon with both n1 and n2 < 10 • 2. Wilcoxon with both n1 and n2≥ 10 • 3. Examples Wilcoxon

  2. Wilcoxon Rank Sum Test • Recall from last week: • When we test a hypothesis about the difference between two independent population means, we do so using the difference between two sample means. • When the two sample variances are tested and found not to be equal • we cannot pool the sample variances • thus we cannot use the t-test for independent samples. Instead, we use the Wilcoxon Rank Sum Test. Wilcoxon

  3. µ tells us about the population X1 X2 The sample mean tells us about µ Population 1 Population 2 µ1 µ2 Sample1 Sample2 Wilcoxon

  4. Wilcoxon Rank Sum Test • The Z test and the t test are “parametric tests” – that is, they answer a question about the difference between populations by comparing sample statistics (e.g., X1 and X2) and making an inference to the population parameters (μ1 and μ2). • The Wilcoxon, in contrast, allows inferences about whole populations Wilcoxon

  5. Distribution A μ X Distribution B μ X Note that distribution B is shifted to the right of distribution A Wilcoxon

  6. 1b. Small samples, independent groups • Wilcoxon Rank Sum Test • first, combine the two samples and rank order all the observations. • smallest number has rank 1, largest number has rank N (= sum of n1 and n2). • separate samples and add up the ranks for the smaller sample. (If n1 = n2, choose either one.) • test statistic : rank sum T for smaller sample. Wilcoxon

  7. 1b. Small samples, independent groups • Wilcoxon – One-tailed Hypotheses • H0: Prob. distributions for 2 sampled populations are identical. • HA: Prob. distribution for Population A shifted to right of distribution for Population B. (Note: could be to the left, but must be one or the other, not both.) Wilcoxon

  8. 1b. Small samples, independent groups • Wilcoxon – Two-tailed Hypotheses • H0: Prob. distributions for 2 sampled populations are identical. • HA: Prob. distribution for Population A shifted to right or left of distribution for Population B. Wilcoxon

  9. 1b. Small samples, independent groups • Wilcoxon – Rejection region: • (With Sample taken from Population A being smaller than sample for Population B) – reject H0 if • TA≥ TU or TA ≤ TL Wilcoxon

  10. 1b. Small samples, independent groups • Wilcoxon for n1≥ 10 and n2 ≥ 10: • Test statistic: • Z = TA – n1(n1 + n2 + 1) • 2 • n1n2(n1 + n2 + 1) • 12 Wilcoxon

  11. Wilcoxon for n1≥ 10 and n2≥ 10 • Rejection region: • One-tailed Two-tailed • Z > Zα │Z│ > Zα/2 • Note: use this only when n1≥ 10 and n2≥ 10 Wilcoxon

  12. Example 1 • These are small samples, and they are independent (“random samples of Cajun and Creole dishes”). Therefore, we have to begin with the test of equality of variances. Wilcoxon

  13. Test of hypothesis of equal variances • H0: 12 = 22 • HA: 12≠ 22 • Test statistic: F = S12 • S22 • Rej. region: F > Fα/2 = F(6,6,.025) = 5.82 • or F < (1/5.82) = .172 Wilcoxon

  14. Test of hypothesis of equal variances • S2Cajun = (385.27)2 = 148432.14 • S2Creole = (1027.54)2 = 1055833.33 • Fobt = 148432.14 = 7.11 • 1055833.33 • Reject H0 – variances are not equal, so we do the Wilcoxon. Wilcoxon

  15. Example 1 – Wilcoxon Rank Sum Test • H0: Prob. distributions for Cajun and Creole populations are identical. • HA: Prob. distribution for Cajun is shifted to right of distribution for Creole. • Statistical test: T Wilcoxon

  16. Example 1 – Wilcoxon Rank Sum Test • Rejection region: • Reject H0 if TCajun > 66 (or if TCreole < 39) • (Note: We shall give lower heat values lower rank values) Wilcoxon

  17. 4.5 11.5 9.5 11.5 4.5 9.5 Example 1 – Wilcoxon Rank Sum Test • Cajun Creole • 3500 3100 • 4200 4700 • 4100 2700 • 4700 3500 • 4200 2000 • 3705 3100 • 4100 1550 6.5 13.5 3 13.5 6.5 2 8 1 Σ 70 35 Wilcoxon

  18. Example 1 – Wilcoxon Rank Sum Test • Calculation check: • Sum of the ranks should = (n) (n+1) • 2 • 70 + 35 = 105 = (14)(15) • 2 Wilcoxon

  19. Example 1 – Wilcoxon Rank Sum Test • TCajun = 70 > 66 (and TCreole = 35 < 39) • Therefore, reject H0 – Cajun dishes are significantly hotter than Creole dishes. Wilcoxon

  20. Example 2 – Wilcoxon Rank Sum Test • H0: 12 = 22 • HA: 12≠ 22 • Test statistic: F = S12 • S22 • Rej. region: F > Fα/2 = F(7,8,.025) = 4.53 • or F < (1/4.90) = .204 Wilcoxon

  21. Example 2 – Wilcoxon Rank Sum Test • Fobt = 4.316 = 9.38 • .46 • Reject H0 – do Wilcoxon Wilcoxon

  22. Example 2 – Wilcoxon Rank Sum Test • H0: Prob. distributions for females and males populations are identical. • HA: Prob. distribution for females is shifted to left of distribution for males. • Statistical test: T • Rejection region: T♂ > TU = 90 • (or T♀ < TL = 54) Wilcoxon

  23. Example 2 – Wilcoxon Rank Sum Test • 6.4 16 2.7 3 • 1.7 1 3.9 10 • 3.2 5 4.6 12 • 5.9 15 3.0 4 • 2.0 2 3.4 6.5 • 3.6 8 4.1 11 • 5.4 14 3.4 6.5 • 7.2 17 4.7 13 • 3.8 9 • Σ 78 75 Wilcoxon

  24. Example 2 – Wilcoxon Rank Sum Test • T♂ = 78 < TU = 90 • Therefore, do not reject H0 – no evidence that mean distance in females is less than that in males. Wilcoxon

  25. Example 3 – Wilcoxon Rank Sum Test • H0: 12 = 22 • HA: 12≠ 22 • Test statistic: F = S12 • S22 • Rej. region: F > Fα/2 = F(5,5,.025) = 7.15 • or F < (1/7.15) = .140 Wilcoxon

  26. Example 3 – Wilcoxon Rank Sum Test • Fobt = (7.563)2 = 57.20 • (2.04)2 4.16 • = 13.74 • Reject H0 – do Wilcoxon Wilcoxon

  27. Example 3 – Wilcoxon Rank Sum Test • H0: Prob. distributions for Hoodoo and Mukluk populations are identical. • HA: Prob. distribution for Hoodoos is shifted to right or left of distribution for Mukluks. • Statistical test: T • Rejection region: TH > 52 or < 26 Wilcoxon

  28. Example 3 – Wilcoxon Rank Sum Test • Hoodoo Mukluk • 2 1 6 5 • 6 5 8 9.5 • 4 2.5 7 7.5 • 23 12 10 11 • 7 7.5 8 9.5 • 6 5 4 2.5 • Σ 33 45 Wilcoxon

  29. Example 3 – Wilcoxon Rank Sum Test • Check: TH + TM = 78 • (12)(13) = 78 • 2 • TH = 33 > 26 and < 52 • Do not reject H0 – no evidence for a significant difference between teams. Wilcoxon

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