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# Chapter 6 Ranksumtest

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### Chapter 6 Ranksumtest

1. 1. 12. Nonparametric test based on ranks
2. 2. <ul><li>A large scale survey reported that the mean of pulses for healthy males is 72 bpm . A physician randomly selected 25 healthy males in a mountainous area and measured their pulses, resulting in a sample mean of 75.2 bpm and a standard deviation of 6.5 bpm . Can one conclude that the mean of pulses for healthy males in the mountainous area is higher than that in the general population （ μ > μ 0 ） ? </li></ul>
3. 3.
4. 4. Supporting Area Rejection Area Rejection Area
5. 5. Parametric Test <ul><li>The methods of hypothesis testing we have learnt </li></ul><ul><li>(1) Assume: the variable follows a normal distribution; </li></ul><ul><li>(2) To test whether the means (parameters) are equal or not under such an assumption. </li></ul><ul><li>Therefore, they are called parametric tests . </li></ul>
6. 6. Non-parametric tests (distribution-free tests) <ul><li>There aren ’t any assumptions about the distribution. </li></ul><ul><li>Chi-square test （ chapter. 6 ） is a kind of non-parametric test. </li></ul><ul><li>Rank sum tests: Another kind of non- </li></ul><ul><li>parametric test, which is based on ranks of the data. </li></ul>
7. 7. <ul><li>Under the following situations, the non-parametric tests could be used: </li></ul><ul><li>a . The distribution of data is unknown; </li></ul><ul><li>b . The distribution of data is skew; </li></ul><ul><li>c . Ranked data or non-precise data; </li></ul><ul><li>d . A quick and brief analysis ( for pilot study ). </li></ul>
8. 8. <ul><li>It is suitable for a variety of data: </li></ul><ul><li>Measurement or enumeration or ordinal </li></ul><ul><li>Normal distribution or not </li></ul><ul><li>Symmetric or not </li></ul><ul><li>However , </li></ul><ul><li>If the data are suitable for parametric tests, </li></ul><ul><li>the power of non-parametric test (if it is used) </li></ul><ul><li>will be slightly lower. </li></ul>
9. 9. 12.1 Wilcoxon’s signed rank sum test (matched pairs) <ul><li>Example 12-1 In order to study the difference of intelligence between twin brothers, the intelligence scores of 12 pairs of twin brothers were measured. The results are listed in Table 12.2. </li></ul>
10. 10. T + =24.5; T - =41.5
11. 11. <ul><li>Steps: </li></ul><ul><li>(1) Hypotheses: </li></ul><ul><li>H 0 : The median of the difference is 0 </li></ul><ul><li>H 1 : The median of the difference is not 0 </li></ul><ul><li>α =0.05. </li></ul><ul><li>(2) Difference </li></ul><ul><li>(3) Ranking absolute differences (omit zero) </li></ul><ul><li>and give back the signs </li></ul><ul><li>(4) Rank sum and statistic </li></ul><ul><li>T = min {positive sum, negative sum} </li></ul><ul><li>(5) P -value and conclusion </li></ul><ul><li>From Table 10 , T is in 10-56, P >0.05, H 0 is not rejected. Conclusion: The intelligence score are at the same level . </li></ul>
12. 12. 12.2 Wilcoxon’s rank sum test for two samples <ul><li>Two independent samples; </li></ul><ul><li>it is not a normal distribution, </li></ul><ul><li>or it is not sure whether the variable </li></ul><ul><li>follows a normal distribution . </li></ul>
13. 13.
14. 14. <ul><li>(1)Hypotheses: </li></ul><ul><li>H 0 : The distributions of two populations are same </li></ul><ul><li>H 1 : The two distributions are not same </li></ul><ul><li>α = 0.05 </li></ul><ul><li>(2) Ranking all the observations in two samples. </li></ul><ul><li>If same values appear in (tie), give a mean rank. </li></ul><ul><li>“ 25” in both sample, and the ranks should be 9 and 10, so that (9+10)/2= 9.5 for each. </li></ul><ul><li>(3) Rank sum for smaller sample, T = T 1 = 78.5 </li></ul><ul><li>(4) P -value and conclusion ( Table 11 ) </li></ul><ul><li>T 0.05,5,9 =28~72, T is outside the range, P <0.05. </li></ul><ul><li>The difference is of statistical significance between two animals. </li></ul>
15. 15. 12.3.1 Kruskal-Wallis’ H test for comparing more than 2 samples <ul><li>Example 12.3 14 newborn infants were grouped into 4 categories according to their mother’s smoking habit. </li></ul><ul><li>A: smoking more than 20 cigarettes per day; </li></ul><ul><li>B: smoking less than 20 cigarettes per day; </li></ul><ul><li>C: ex-smoker; </li></ul><ul><li>D: never smoking. </li></ul><ul><li>Their weights are listed in Table 12.7. </li></ul>
16. 16.
17. 17. <ul><li>(1)Hypothesis : </li></ul><ul><li>H 0 : The distributions of three populations are all same </li></ul><ul><li>H 1 : The distributions of three populations are not all same </li></ul><ul><li>α = 0.05 </li></ul><ul><li>(2) Ranking all the observations in three samples </li></ul><ul><li>(Same way for ties) </li></ul><ul><li>(3) Rank sums for each sample </li></ul><ul><li>R 1 = R 2 =15, R 3 = R 4 =37.5 </li></ul>
18. 18. <ul><li>(4) Statistic H </li></ul><ul><li>If there is no tie </li></ul><ul><li>If there are ties </li></ul><ul><li>t j : Number of individuals in j - th tie </li></ul><ul><li>Example 12.7: </li></ul>
19. 19. (5) P -value and conclusion —— Compare with critical value of H ( C 7 ) or k : Number of samples Example 12.7: Conclusion: The weights are not all at an equal level.
20. 20. 12.3.2 Friedman test for the data from a randomized block design Example 12.4 The riboflavin were tested for 3 samples of cabbage under four test conditions (A, B, C and D). The results are listed in Table 12.9. Now the question is if the test results are different in different kinds of test conditions.
21. 21.
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24. 24. 12.3.3. multiple comparison of mean ranks <ul><li>When the comparison among four groups results in </li></ul><ul><li>significant differences, multiple comparison is needed to </li></ul><ul><li>know who and who are different. Z tests for pair-wise </li></ul><ul><li>comparison could be used. </li></ul><ul><li>H 0 : The location of population A and B are different </li></ul><ul><li>H 1 : The location of population A and B are not different </li></ul><ul><li>α = 0.05 </li></ul>
25. 25.
26. 26. <ul><li>(1)Hypothesis: </li></ul><ul><li>H 0 : this pair of two population distributions have the same location </li></ul><ul><li>H 1 : this pair of two population distributions have different locations, </li></ul><ul><li>α =0.05. </li></ul><ul><li>(2) Calculate Z value: </li></ul>
27. 27. <ul><li>(3) Decide P value , </li></ul><ul><li>Weights in first group has a different level from that of fourth group. Since , The mothers who smoke may have babies with lower weights. </li></ul>Conclusion: Smoking may lead to the newborn ’ s lower weights.
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