1 / 14

Chapter 15

Chapter 15. Nonparametric Methods. Nonparametric Methods. 15.1 The Sign Test: A Hypothesis Test about the Median 15.2 The Wilcoxon Rank Sum Test 15.3 The Wilcoxon Signed Ranks Test 15.4 Comparing Several Populations Using the Kruskal-Wallis H Test

hertz
Télécharger la présentation

Chapter 15

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 15 Nonparametric Methods

  2. Nonparametric Methods 15.1 The Sign Test: A Hypothesis Test about the Median 15.2 The Wilcoxon Rank Sum Test 15.3 The Wilcoxon Signed Ranks Test 15.4 Comparing Several Populations Using the Kruskal-Wallis H Test 15.5 Spearman’s Rank Correlation Coefficient

  3. Sign Test: A Hypothesis Test aboutthe Median Alternative Test Statistic p-Value Define S = the number of sample measurements (less/greater) than M0, x to be a binomial random variable with p = 0.5 We can reject H0: Md = M0 at the  level of significance (probability of Type I error equal to ) by using the appropriate p-value

  4. The Large Sample Sign Test for aPopulation Median Reject H0 if: p-Value Alternative Test Statistic If the sample size n is large (n  10), we can reject H0: Md = M0at the  level of significance (probability of Type I error equal to ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than 

  5. The Wilcoxon Rank Sum Test • Given two independent samples of sizes n1 and n2 from populations 1 and 2 with distributions D1 and D2 • Rank the (n1+ n2) observations from smallest to largest (average ranks for ties) • T1 = sum of ranks, sample 1 • T2 = sum of ranks, sample 2 • T = T1 if n1 n2 and T = T2 if n1> n2 • We can reject H0: D1 and D2 are identical probability distributions at the  level of significance if and only if the test statistic T satisfies the appropriate rejection point condition

  6. The Wilcoxon Rank Sum Test Continued Reject H0 if: Alternative TU and TL are given for n1 and n2 between 3 and 10 in Table A.15

  7. The Large Sample Wilcoxon RankSum Test • Given two large (n1, n210) independent samples from populations 1 and 2 with distributions D1 and D2 • Rank the (n1+ n2) observations from smallest to largest (average ranks for ties) • Let T = T1 = sum of ranks, sample 1 • We can reject H0: D1 and D2 are identical probability distributionsat the  level of significance if and only if the test statistic z satisfies the appropriate rejection point condition or, equivalently, if the corresponding p-value is less than 

  8. The Large Sample Wilcoxon RankSum Test Continued p-Value Alternative Reject H0 if: Test Statistic

  9. The Wilcoxon Signed Rank Test • Given two matched pairs of n observations, selected at random from populations 1 and 2 with distributions D1 and D2 • Compute the n differences (D1 – D2) • Rank the absolute value of the differences from smallest to largest • Drop zero differences from sample • Assign average ranks for ties • T- = sum of ranks, negative differences • T+ = sum of ranks, positive differences • We can reject H0: D1 and D2 are identical probability distributions at the  level of significance if and only if the appropriate test statistic satisfies the corresponding rejection point condition

  10. The Wilcoxon Signed Rank TestContinued Alternative Test Statistic Reject H0 if: Rejection points T0 are given for n between 5 and 50 in Table A.16

  11. The Large Sample Wilcoxon SignedRank Test • Given two large samples (n  10) of matched pairs of observations from populations 1 and 2 with distributions D1 and D2 • Compute the n differences (D1 – D2) • Rank the absolute value of the differences from smallest to largest • Drop zero differences from sample • Assign average ranks for ties • Let T = T+ = sum of ranks, positive differences • We can reject H0: D1 and D2 are identical probability distributions at the  level of significance if and only if the test statistic z satisfies the appropriate rejection point condition or, equivalently, if the corresponding p-value is less than 

  12. The Large Sample Wilcoxon SignedRank Test Continued Alternative Reject H0 if: p-Value Test Statistic

  13. The Kruskal-Wallis H Test H0: The p populations are identical Ha: At least two of the populations differ in location To Test: Given p independent samples (n1, …, np 5) from p populations. Rank the (n1+ … + np) observations from smallest to largest (average ranks for ties.) Let T1 = sum of ranks, sample 1; …; Tp = sum of ranks, sample p Test Statistic: Reject H0 if H > a2or if p-value < a a2is based on p-1 degrees of freedom

  14. Spearman’s Rank CorrelationCoefficient Given n pairs of measurements on two variables, x and y, rank the values of x and y separately, assigning average ranks in case of ties Then the Spearman rank correlation coefficient, rs is given by the standard Pearson correlation coefficient (Section 11.6) of the ranks. If there are no ties in the ranks, the Spearman correlation coefficient can be calculated as Where di is the difference between the x-rank and the y-rank for the ith observation

More Related