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# Hph7300week14winter2009narr

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### Hph7300week14winter2009narr

1. 1. Ordinal Dependent Variable or Non-normal Continuous Variable Week 14 4/6/09 Comparing non-parametric statistics Final due 4/12/09
2. 2. Wilcoxon Rank-Sum Test for Differences in 2 Medians <ul><li>Test two independent population medians </li></ul><ul><li>Populations need not be normally distributed </li></ul><ul><li>Distribution free procedure </li></ul><ul><li>Used when only rank data is available </li></ul>
3. 3. Wilcoxon Rank-Sum Test <ul><li>For large samples, the test statistic T 1 is approximately normal with mean and standard deviation : </li></ul><ul><ul><li>Must use the normal approximation if either n 1 </li></ul></ul><ul><ul><li> or n 2 > 10 </li></ul></ul><ul><ul><li>Can use the normal approximation for small samples </li></ul></ul><ul><ul><li>Assign n 1 to be the smaller of the two sample sizes </li></ul></ul>
4. 4. Wilcoxon Rank-Sum Test <ul><li>The Z test statistic is </li></ul><ul><li>Where Z approximately follows a standardized normal distribution </li></ul>
5. 5. <ul><li>Sample data is collected on the capacity rates (% of capacity) for two emergency rooms. </li></ul><ul><li>Are the median operating rates for two ER’s the same? </li></ul><ul><li>For Hospital A , the rates are 71, 82, 77, 94, 88 </li></ul><ul><li>For Hospital B, the rates are 85, 82, 92, 97 </li></ul><ul><li>Test for equality of the sample medians at the 0.05 significance level </li></ul>Wilcoxon Rank-Sum Test:
6. 6. Wilcoxon Rank-Sum Test: Tie in 3 rd and 4 th places Ranked Capacity values: Capacity Rank Hosp A Hosp B Hosp A Hosp B 71 1 77 2 82 3.5 82 3.5 85 5 88 6 92 7 94 8 97 9 Rank Sums: 20.5 24.5
7. 7. Wilcoxon Rank-Sum Test: <ul><li>The level of significance was α = .05 </li></ul><ul><li>The test statistic was T 1 = 24.5 </li></ul>
8. 8. Wilcoxon Rank-Sum Test: <ul><li>The test statistic is </li></ul><ul><li>Z = 1.64 is not greater than the critical Z value of 1.96 (for α = .05) so we do not reject H 0 – there is not sufficient evidence that the medians are not equal </li></ul>
9. 9. Kruskal-Wallis Rank Test for c Medians <ul><li>Used to analyze completely randomized experimental designs </li></ul><ul><li>Use  2 distribution to approximate if each sample group size n j > 5 </li></ul><ul><ul><li>df = c – 1 </li></ul></ul>
10. 10. Kruskal-Wallis Rank Test <ul><li>Assumptions </li></ul><ul><ul><li>Independent random samples are drawn </li></ul></ul><ul><ul><li>Continuous dependent variable </li></ul></ul><ul><ul><li>Data may be ranked both within and among samples </li></ul></ul><ul><ul><li>Populations have same variability </li></ul></ul><ul><ul><li>Populations have same shape </li></ul></ul><ul><li>Robust with regard to last two conditions </li></ul><ul><ul><li>Use F test in completely randomized designs and when the more stringent assumptions hold </li></ul></ul>
11. 11. Kruskal-Wallis Rank Test: <ul><li>As pharmacy manager, you want to see if three filling machines have different median filling times. You assign 15 similarly trained & experienced workers, five per machine, to the machines. At the .05 significance level, is there a difference in median filling times? </li></ul>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
12. 12. Obtaining a Ranking Machine1 Machine2 Machine3 14 9 2 15 6 7 12 10 1 11 8 4 13 5 3 Raw Data Ranks 65 38 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
13. 13. Kruskal-Wallis Test Procedure <ul><li>The Kruskal-Wallis H test statistic: </li></ul><ul><li> (with c – 1 degrees of freedom) </li></ul>where: n = sum of sample sizes in all samples c = Number of samples T j = Sum of ranks in the j th sample n j = Size of the j th sample
14. 14. Kruskal-Wallis Test Example Solution <ul><li>H 0 : M 1 = M 2 = M 3 </li></ul><ul><li>H 1 : Not all equal </li></ul><ul><li> = .05 </li></ul><ul><li>df = c - 1 = 3 - 1 = 2 </li></ul><ul><li>Critical Value(s): </li></ul>Decision: Conclusion:  = .05 H = 11.58
15. 15. Nonparametric Equivalents <ul><li>Z-test or t-test for independent samples </li></ul><ul><ul><li>Wilcoxon Rank Sum test </li></ul></ul><ul><ul><li>Mann Whitney U test </li></ul></ul><ul><li>Z-test or t-test for dependent samples </li></ul><ul><ul><li>Wilcoxon Sign Rank test </li></ul></ul><ul><li>ANOVA </li></ul><ul><ul><li>Kruskal-Wallis test </li></ul></ul><ul><li>Repeated Measures ANOVA </li></ul><ul><ul><li>Friedman test </li></ul></ul>