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Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
Chapter 6  Ranksumtest
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Chapter 6 Ranksumtest

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  • 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.

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