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# Chapter 6 Ranksumtest

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### Transcript

• 1. 12. Nonparametric test based on ranks
• 2.
• 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 ） ?
• 3.
• 4. Supporting Area Rejection Area Rejection Area
• 5. Parametric Test
• The methods of hypothesis testing we have learnt
• (1) Assume: the variable follows a normal distribution;
• (2) To test whether the means (parameters) are equal or not under such an assumption.
• Therefore, they are called parametric tests .
• 6. Non-parametric tests (distribution-free tests)
• There aren ’t any assumptions about the distribution.
• Chi-square test （ chapter. 6 ） is a kind of non-parametric test.
• Rank sum tests: Another kind of non-
• parametric test, which is based on ranks of the data.
• 7.
• Under the following situations, the non-parametric tests could be used:
• a . The distribution of data is unknown;
• b . The distribution of data is skew;
• c . Ranked data or non-precise data;
• d . A quick and brief analysis ( for pilot study ).
• 8.
• It is suitable for a variety of data:
• Measurement or enumeration or ordinal
• Normal distribution or not
• Symmetric or not
• However ,
• If the data are suitable for parametric tests,
• the power of non-parametric test (if it is used)
• will be slightly lower.
• 9. 12.1 Wilcoxon’s signed rank sum test (matched pairs)
• 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.
• 10. T + =24.5; T - =41.5
• 11.
• Steps:
• (1) Hypotheses:
• H 0 : The median of the difference is 0
• H 1 : The median of the difference is not 0
• α =0.05.
• (2) Difference
• (3) Ranking absolute differences (omit zero)
• and give back the signs
• (4) Rank sum and statistic
• T = min {positive sum, negative sum}
• (5) P -value and conclusion
• 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 .
• 12. 12.2 Wilcoxon’s rank sum test for two samples
• Two independent samples;
• it is not a normal distribution,
• or it is not sure whether the variable
• follows a normal distribution .
• 13.
• 14.
• (1)Hypotheses:
• H 0 : The distributions of two populations are same
• H 1 : The two distributions are not same
• α = 0.05
• (2) Ranking all the observations in two samples.
• If same values appear in (tie), give a mean rank.
• “ 25” in both sample, and the ranks should be 9 and 10, so that (9+10)/2= 9.5 for each.
• (3) Rank sum for smaller sample, T = T 1 = 78.5
• (4) P -value and conclusion ( Table 11 )
• T 0.05,5,9 =28~72, T is outside the range, P <0.05.
• The difference is of statistical significance between two animals.
• 15. 12.3.1 Kruskal-Wallis’ H test for comparing more than 2 samples
• Example 12.3 14 newborn infants were grouped into 4 categories according to their mother’s smoking habit.
• A: smoking more than 20 cigarettes per day;
• B: smoking less than 20 cigarettes per day;
• C: ex-smoker;
• D: never smoking.
• Their weights are listed in Table 12.7.
• 16.
• 17.
• (1)Hypothesis :
• H 0 : The distributions of three populations are all same
• H 1 : The distributions of three populations are not all same
• α = 0.05
• (2) Ranking all the observations in three samples
• (Same way for ties)
• (3) Rank sums for each sample
• R 1 = R 2 =15, R 3 = R 4 =37.5
• 18.
• (4) Statistic H
• If there is no tie
• If there are ties
• t j : Number of individuals in j - th tie
• Example 12.7:
• 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. 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.
• 22.
• 23.
• 24. 12.3.3. multiple comparison of mean ranks
• When the comparison among four groups results in
• significant differences, multiple comparison is needed to
• know who and who are different. Z tests for pair-wise
• comparison could be used.
• H 0 : The location of population A and B are different
• H 1 : The location of population A and B are not different
• α = 0.05
• 25.
• 26.
• (1)Hypothesis:
• H 0 : this pair of two population distributions have the same location
• H 1 : this pair of two population distributions have different locations,
• α =0.05.
• (2) Calculate Z value:
• 27.
• (3) Decide P value ,
• Weights in first group has a different level from that of fourth group. Since , The mothers who smoke may have babies with lower weights.
Conclusion: Smoking may lead to the newborn ’ s lower weights.