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15-2 Nonparametric Methods 15.1The Sign Test: A Hypothesis Test about the MedianThe Sign Test: A Hypothesis Test about the Median 15.2The Wilcoxon Rank Sum TestThe Wilcoxon Rank Sum Test 15.3The Wilcoxon Signed Ranks TestThe Wilcoxon Signed Ranks Test 15.4Comparing Several Populations Using the Kruskal-Wallis H TestComparing Several Populations Using the Kruskal-Wallis H Test 15.5Spearman’s Rank Correlation CoefficientSpearman’s Rank Correlation Coefficient

15-3 Sign Test: A Hypothesis Test about the Median Define S = the number of sample measurements (less/greater) than M 0, x to be a binomial random variable with p = 0.5 We can reject H 0 : M d = M 0 at the  level of significance (probability of Type I error equal to  ) by using the appropriate p-value Alternativep-ValueTest Statistic

15-4 The Large Sample Sign Test for a Population Median Test Statistic If the sample size n is large (n  10), we can reject H 0 : M d = M 0 at 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  Alternative Reject H 0 if:p-Value

15-5 The Wilcoxon Rank Sum Test Given two independent samples of sizes n 1 and n 2 from populations 1 and 2 with distributions D 1 and D 2 Rank the (n 1 + n 2 ) observations from smallest to largest (average ranks for ties) T 1 = sum of ranks, sample 1 T 2 = sum of ranks, sample 2 T = T 1 if n 1  n 2 and T = T 2 if n 1 > n 2 We can reject H 0 : D 1 and D 2 are identical probability distributions at the  level of significance if and only if the test statistic T satisfies the appropriate rejection point condition

15-6 The Wilcoxon Rank Sum Test Continued Alternative Reject H 0 if: T U and T L are given for n 1 and n 2 between 3 and 10 in Table A.15

15-7 The Large Sample Wilcoxon Rank Sum Test Given two large (n 1, n 2  10) independent samples from populations 1 and 2 with distributions D 1 and D 2 Rank the (n 1 + n 2 ) observations from smallest to largest (average ranks for ties) Let T = T 1 = sum of ranks, sample 1 We can reject H 0 : D 1 and D 2 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 

15-8 The Large Sample Wilcoxon Rank Sum Test Continued Test Statistic AlternativeReject H 0 if: p-Value

15-9 The Wilcoxon Signed Rank Test Given two matched pairs of n observations, selected at random from populations 1 and 2 with distributions D 1 and D 2 Compute the n differences (D 1 – D 2 ) 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 H 0 : D 1 and D 2 are identical probability distributions at the  level of significance if and only if the appropriate test statistic satisfies the corresponding rejection point condition

15-10 The Wilcoxon Signed Rank Test Continued Rejection points T 0 are given for n between 5 and 50 in Table A.16 Alternative Reject H 0 if: Test Statistic

15-11 The Large Sample Wilcoxon Signed Rank Test Given two large samples (n  10) of matched pairs of observations from populations 1 and 2 with distributions D 1 and D 2 Compute the n differences (D 1 – D 2 ) 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 H 0 : D 1 and D 2 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 

15-12 The Large Sample Wilcoxon Signed Rank Test Continued Test Statistic AlternativeReject H 0 if:p-Value

15-13 The Kruskal-Wallis H Test Test Statistic: Reject H 0 if H >     or if p-value <      is based on p-1 degrees of freedom Given p independent samples (n 1, …, n p  5) from p populations. Rank the (n 1 + … + n p ) observations from smallest to largest (average ranks for ties.) Let T 1 = sum of ranks, sample 1; …; T p = sum of ranks, sample p H 0 : The p populations are identical H a : At least two of the populations differ in location To Test:

15-14 Spearman’s Rank Correlation Coefficient 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, r s 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 d i is the difference between the x-rank and the y-rank for the i th observation