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In previous testing, we assumed that our samples were drawnfrom normally distributed populations.This chapter introduces some techniques that do not makethat assumption.These methods are called distribution-free or nonparametrictests.In situations where the normal assumption is appropriate,nonparametric tests are less efficient than traditionalparametric methods.Nonparametric tests frequently make use only of the order ofthe observations and not the actual values.
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In this section, we will discuss four nonparametric tests:the Wilcoxon Rank Sum Test (or Mann-Whitney U test),the Wilcoxon Signed Ranks Test,the Kruskal-Wallis Test, andthe one sample test of runs.
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The Wilcoxon Rank Sum Test or Mann-Whitney U TestThis test is used to test whether 2 independent samples havebeen drawn from populations with the same median.It is a nonparametric substitute for the t-test on the differencebetween two means.
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Wilcoxon Rank Sum Test Example: university Based on the following samples fromA B two universities, test at the 10% level50 70 whether graduates from the two52 73 schools have the same average grade56 77 on an aptitude test.60 8064 8368 8571 8774 8889 9695 99
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First merge and rank the grades. rank grade universitySum the ranks for each sample. 1 50 Arank sum for university A: 74 2 52 Arank sum for university B: 136 3 56 A 4 60 A university 5 64 A 6 68 A Note: If there are A B 7 70 B ties, each value 50 70 8 71 A gets the average 52 73 9 73 B rank. For example, 10 74 A if 2 values tie for 56 77 11 77 B 3th and 4th place, 60 80 both are ranked 12 80 B 64 83 13 83 B 3.5. If three differences would 68 85 14 85 B 15 87 B be ranked 7, 8, and 71 87 9, rank them all 8. 16 88 B 74 88 17 89 A 89 96 18 95 A 19 96 B 95 99 20 99 B
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Here, the group from university A is considered the 1st sample. When the samples differ in size, designate the smaller of the 2 samples as the 1st sample. Define T1 = sum of the ranks for 1st sample . n1 (n1 + n2 + 1) The mean of T1 is µT1 = , 2 n1n 2 (n1 + n2 + 1) and the standard deviation is σ T1 = . 12 If n1 and n 2 are each at least 10, T1 is approximately normal. T1 - µT1 So, Z = has a standard normal distribution. σ T1(For small sample sizes, the Z approximation is sometimes used as well.)
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For our example, T1 = 74. n1 (n + 1) 10(20 + 1) µT1 = = = 105 2 2 n1n 2 (n + 1) (10)(10)(20 + 1) σ T1 = = = 13.229 12 12 T1 - µT1 74 - 105 Z = = = -2.343. σ T1 13.229Since the critical values for a2-tailed Z test at the 10%level are 1.645 and -1.645, we critical critical region regionreject H0 that the medians are .45 .45the same and accept H1 that .05 .05the medians are different. -1.645 0 1.645 Z
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For small sample sizes, you can use Table E.8 inyour textbook, which provides the lower and uppercritical values for the Wilcoxon Rank Sum Test.That table shows that for our 10% 2-tailed test, the lower critical value is 82 and the upper critical value is 128.Since our smaller sample’s rank sum is 74, which is outside the interval (82, 128) indicated in the table, we reject the null hypothesis that the medians are the same and conclude that they are different.Equivalently, since the larger sample’s rank sum is 136, which is also outside the interval (82, 128), we again reject the null hypothesis that the medians are the same and conclude that they are different.
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The Wilcoxon Signed Rank TestThis test is used to test whether 2 dependent samples havebeen drawn from populations with the same median.It is a nonparametric substitute for the paired t-test on thedifference between two means.
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Wilcoxon Signed Rank Test Procedure1. Calculate the differences in the paired values (Di=X1i – X2i)2. Take absolute values of the differences and rank them (Discard all differences that equal 0.)3. Assign ranks Ri with the smallest rank equal to 1. As in the rank sum test, if two or more of the differences are equal, each difference gets the average rank. (That is, if two differences would be ranked 3 and 4, rank them both 3.5. If three differences would be ranked 7, 8, and 9, rank them all 8.)4. Assign the symbol + to positive differences and – to negative differences.5. Calculate the Wilcoxon statistic W as the sum of the positive ranks. So, W= ∑ Ri+
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Wilcoxon Signed Rank Test Procedure (cont’d)In the following, n refers to the number of non - zero differences. n(n + 1) The mean of the Wilcoxon statistic W is µW = 4 The standard deviation of the Wilcoxon statistic W is n(n + 1)(2n + 1) σW = 24If n is at least 20, the test statistic W is approximately normal. So we have : W − µW Z= σW(For small sample sizes, the Z approximation is sometimes used as well.)
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diff rank rank diff rank rank exam1 exam2 exam1 exam2 (ex2-ex1) (+) (-) (ex2-ex1) (+) (-) Example 95 97 72 68Suppose we have 76 76 78 94a class with 22 82 75 58 55students, each ofwhom has two 48 54 73 75exam grades. 27 31 71 70We want to test at 34 39 69 66the 5% level 58 61 57 62whether there is adifference in the 98 97 84 92median grade for 45 45 91 81the two exams. 77 94 83 90 27 36 67 73
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diff rank rank diff rank rank exam1 exam2 exam1 exam2 (ex2-ex1) (+) (-) (ex2-ex1) (+) (-)Then we total the 95 97 2 3.5 72 68 -4 8.5signed ranks. We 76 76 0 78 94 16 19get 154 for the sumof the positive ranks 82 75 -7 14.5 58 55 -3 6and 56 for the sum of 48 54 6 12.5 73 75 2 3.5the negative ranks.The Wilcoxon test 27 31 4 8.5 71 70 -1 1.5statistic is the sum of 34 39 5 10.5 69 66 -3 6the positive ranks.So W = 154. 58 61 3 6 57 62 5 10.5 98 97 -1 1.5 84 92 8 16 45 45 0 91 81 -10 18 77 94 17 20 83 90 7 14.5 27 36 9 17 67 73 6 12.5 154 56
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Since we had 22 students and 2 zero differences, the number of non-zero differences n = 20. n(n + 1) (20)(21) Recall that the mean of W is µW = = = 105 4 4 The standard deviation of W is n(n + 1)(2n + 1) 20(21)(41) σW = = = 26.786 24 24 W − µW 154 − 105So we have : Z= = = 1.829 σW 26.786Since the critical values for a2-tailed Z test at the 5% level critical criticalare 1.96 and -1.96, we can not region .475 .475 regionreject the null hypothesis H0 and .025 .025so we conclude that the mediansare the same. -1.96 0 1.96 Z
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For small sample sizes, you can use Table E.9 inyour textbook, which provides the lower and uppercritical values for the Wilcoxon Signed Rank Test.That table shows that for our 5% 2-tailed test, the lower critical value is 52 and the upper critical value is 158.Since the sum of our positive ranks is 154, which is inside the interval (52, 158) indicated in the table, we can not reject the null hypothesis and so we conclude that the medians are the same.
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The Kruskal-Wallis TestThis test is used to test whether several populations have thesame median.It is a nonparametric substitute for a one-factor ANOVA F-test.
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12 R j 2The test statistic is K = ∑ - 3(n + 1) , n(n + 1) nj where nj is the number of observations in the jth sample, n is the total number of observations, and Rj is the sum of ranks for the jth sample.If each n j ≥ 5 and the null hypothesis is true,then the distribution of K is χ 2 with dof = c - 1,where c is the number of sample groups.In the case of ties, a corrected statistic should be computed: K Kc = where tj is the number of ties in ∑ (t 3 − t j ) j 1- the jth sample. n −n 3
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Kruskal-Wallis Test Example: Test at the 5% level whetheraverage employee performance is the same at 3 firms, usingthe following standardized test scores for 20 employees. Firm 1 Firm 2 Firm 3 score rank score rank score rank 78 68 82 95 77 65 85 84 50 87 61 93 75 62 70 90 72 60 80 73 n1 = 7 n2 = 6 n3 =7
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f(χ2) crit. acceptance reg. region .05 5.991 χ 22From the χ2 table, we see that the 5% critical value for a χ2with 2 dof is 5.991.Since our value for K was 6.641, we reject H0 that themedians are the same and accept H1 that the medians aredifferent.
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One sample test of runsa test for randomness of order of occurrence
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A run is a sequence of identical occurrencesthat are followed and preceded by differentoccurrences.Example: The list of X’s & O’s below consists of 7 runs.xxxooooxxooooxxxxoox
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Suppose r is the number of runs, n1 is the number oftype 1 occurrences and n2 is the number of type 2occurrences.The mean number of runs is 2n1n 2 μr = + 1. n1 + n 2The standard deviation of the number of runs is 2n1n 2 (2n1n 2 - n1 - n 2 ) σr = . (n1 + n 2 ) (n1 + n 2 − 1) 2
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If n1 and n2 are each at least 10, then r isapproximately normal. r - µr So, Z= σr is a standard normal variable.
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Example: A stock exhibits the following price increase (+) and decrease (−) behavior over 25 business days. Test at the 1% whether the pattern is random. r =16, + + + − − + − − − + + − + − + − − + + − + + − + − n1 (+) = 13, 2n1n 2 n2 (−) = 2(13)(12) μr = +1 = + 1 = 13.48 12 n1 + n 2 13 + 12 2n1n 2 (2n1n 2 - n1 - n 2 ) 2(13)(12) [(2(13)(12) - 13 - 12]σr = = = 2.44 (n1 + n 2 ) (n1 + n 2 − 1) 2 (13 + 12) (13 + 12 − 1) 2 r - µ r 16 - 13.48 Z= = = 1.03 critical critical σr 2.44 region .495 .495 acceptance region .005 .005 Since the critical values for a 2-tailed 1% region test are 2.575 and -2.575, we accept H0 -2.575 0 2.575 Z that the pattern is random.
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